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"Babylonian magic is a lot more complex than its Sumerian counterpart and incorporates several new techniques, making it a lot harder to succinctly describe. The two biggest additions, in my opinion, are the introduction of the magical ceremony, and a reliance on prognostication for insight into the will of the Gods.
Prognostication is a system for decoding ominous phenomena to extract personal meaning. Today we would call this divination, and the Babylonians practiced many different forms, including: astrology (interpreting ominious celestial and atmospheric phenomena), belomancy (using arrows as tools for divining answers to questions), extaspicy (consulting the entrails of sacrificial animals for answers to questions), and lecanomancy (looking for significance in the patterns formed by oil spilled into bowls of water).
The word for a diviner in Babylonian is bārû, and the corpus of material they were expected to learn and consult while plying their trade is referred to as bārûtu. This corpus often comes to us in the form of an iškāru, a series of tablets outlining the relevant material for a given subject. Several of these series have survived, albeit in in incomplete forms. Three of them appear to have been the primary focus of the Babylonian bārû:
Šumma ālu ina mēlê šakin, consisting of more than 100 tablets, covers various ominous phenomena related to city planning, house construction, supernatural encounters with ghosts and daemons, the weather, and the behavior of humans and animals encountered in everyday life.
Enūma Anu Enlil, consisting of 68 to 70 tablets, details atmospheric and celestial omens and their interpretation. Entire sections of this series focus on things like thunderstorms, where certain planets are in the nighttime sky in relation to the asterisms recognized by the Babylonians, and other celestial events that can be observed from the earth, such as comets, halos around the Moon, clouds, etc.
Šumma immeru deals exclusively with the behavior of sacrificial sheep prior to their ritual slaughter, and the interpretation of their exta in the aftermath. This series also includes an overview of the ritual that Babylonian bārû were expected to perform in order to transform a sacrificial sheep from normal livestock into a vessel through which the divine will could be transmitted.
As of today, various universities have on-going volumes where transliterations, translations, and commentary for each of these series are being published. However, these are not books for the layman and aren't usually found in popular bookstores or occult forums. They are, for all intents and purposes, academic textbooks intended to present the information in as unfiltered of a form as it can be. Simply buying one of these books won't actually enable you to perform the divination the way that owning a copy of The Key of Solomon would.
Here's a small selection of titles focused on prognostication:
Cohen, Yoram. 2020.The Babylonian šumma immeru Omens: Transmission, Reception, and Text Production (Dubsar 9). Münster, Germany: Zaphon.
Freedman, Sally M. 1998. If A City is Set on A Height. The Akkadian Omen Series Šumma Alu ina Mēlê Šakin. Volume 1: Tablets 1-21. Occasional Publications of the Samuel Noah Kramer Fund, 17. Philadelphia, PA: University of Pennsylvania Museum.
Freedman, Sally M. 2006. If A City is Set on A Height. The Akkadian Omen Series Šumma Alu ina Mēlê Šakin. Volume 2: Tablets 22-40. Occasional Publications of the Samuel Noah Kramer Fund, 19. Philadelphia, PA: University of Pennsylvania Museum.
Freedman, Sally M. 2017. If A City is Set on A Height. The Akkadian Omen Series Šumma Alu ina Mēlê Šakin. Volume 3: Tablets 41-63. Winona Lake, IN: Eisenbrauns.
Gehlken, Erlend. 2012. Weather Omens of Enūma Anu Enlil: Thunderstorms, Wind and Rain (Tablets 44-49). Cuneiform Monographs, Vol. 43. Leiden, Netherlands: BRILL.
Koch-Westenholz, Ulla. 1995. Mesopotamian Astrology: An Introduction to Babylonian and Assyrian Celestial Divination. Carsten Niebuhr Institute Publications, No. 19. Copenhagen, Denmark: Carsten Niebuhr Institute of Near Eastern Studies: Museum Tusculanum Press, University of Copenhagen.
Oppenheim, A. Leo. 1956. The Interpretation of Dreams in the Ancient Near East, With A Translation of An Assyrian Dream Book. Transactions of the American Philosophical Society, Vol. 46/3. Philadelphia, PA: American Philosophical Society. Reiner, Erica. 2005. Babylonian Planetary Omens Part 4. Cuneiform Monographs, Vol. 30. Leiden, Netherlands: BRILL.
Reiner, Erica & David Pingree. 1975. Babylonian Planetary Omens Part 1: The Venus Tablet of Ammiṣaduqa. (Bibliotheca Mesopotamica). California: Undena Publications.
Reiner, Erica & David Pingree. 1981. Babylonian Planetary Omens Part 2: Enūma Anu Enlil, Tablet 50-51. (Bibliotheca Mesopotamica). California: Undena Publications.
Reiner, Erica & David Pingree. 1998. Babylonian Planetary Omens Part 3. Cuneiform Monographs, Vol. 11. Leiden, Netherlands: BRILL.
Rochberg, Fransesca. 2004. The Heavenly Writing: Divination, Horoscopy, and Astronomy in Mesopotamian Culture. New York, NY: Cambridge University Press.
Rochberg, Francesca. 2010. In the Path of the Moon: Babylonian Celestial Divination and Its Legacy. Ancient Magic and Divination, Vol. 6. Leiden, Netherlands: BRILL." source

The second major innovation is the magical ceremony. Babylonian magicians went by various titles, the significance of which is not perfectly understood in the modern day. Two of the more common words for Babylonian magicians are āšipu and mašmaššu.
As with the diviner discussed above, the Babylonian magician also had a corpus of material at his or her disposal, referred to as āšipūtu. Like the diviner, this corpus consisted of various iškāru series, although rather than lists of ominous phenomena and their interpretation, the magical series were compendiums of exorcisms, incantations, and prayers to be recited by the magician or their patient. The final tablet in each series is called its ṭuppi nēpēši, literally "procedural tablet," and includes all of the nonverbal actions the magician is supposed to perform during the ceremony.
Of course, these series are, at their heart, composed of individual exorcisms, incantations, and prayers. Many of these were no doubt originally independent of the series despite primarily coming down to us as part of the compendium. Additionally, there are dozens of different classifications for each type of ritual, many of which identify its overall genre (such as a "cultic song" or an "apotropaic incantation") or specific purpose (a šuʾila, "lifted hand," prayer for intercession; a namburbû, "apotropaeon," being a prophylactic ritual; etc.).
The āšipu and their āšipūtu are clearly an evolution on the older Sumerian magician and his legend-incantations, to the point where the magical formula I mentioned in my original reply—šiptum ul yattun šipat DN, "The incantation is not mine, it is the incantation of deity-X!"—appears regularly throughout the various Babylonian series that have come down to us, as well as the driving force behind these exorcisms, incantations, and prayers continuing to be "theistic operative force," i.e., a deity—or, more properly, a pair or group of deities—empowers the magician to perform his or her rituals.
Due to the longevity of the Babylonian Empire (ca. 1900–539 BCE), we have a significantly larger corpus of material to work with when it comes to Babylonian exorcisms, incantations, and prayers. This means there are also numerous volumes available that present transliterations, translations, and commentary on both series and individual tablets with examples of magical works. As with the prognostication series though, these volumes are, again, presented by academics primarily for other academics. If you don't have a vested interest in studying and learning authentic Babylonian magical techniques, the books listed below won't be of much use.
Abusch, Tzvi. 2016. The Magical Ceremony Maqlû: A Critical Edition. Ancient Magic and Divination, Vol. 10. Leiden, Netherlands: BRILL.
Abusch, Tzvi & Schwemmer, Daniel. 2011. Corpus of Mesopotamian Anti-Witchcraft Rituals (Vol. 1). Ancient Magic and Divination, Vol. 8/1. Leiden, Netherlands: BRILL.

Abusch, Tzvi & Schwemmer, Daniel. 2016. Corpus of Mesopotamian Anti-Witchcraft Rituals (Vol. 2). Ancient Magic and Divination, Vol. 8/2. Leiden, Netherlands: BRILL.
Abusch, Tzvi; Schwemmer, Daniel; Luukko, Mikko; & Van Buylaere, Greta. 2019. Corpus of Mesopotamian Anti-Witchcraft Rituals (Vol. 3). Ancient Magic and Divination, Vol. 8/3. Leiden, Netherlands: BRILL.
Farber, Walter. 2014. Lamaštu: An Edition of the Canonical Series of Lamaštu Incantations and Rituals and Related Texts from the Second Millennium BC. Winona Lake, IN: Eisenbrauns.
Geller, Markham J. 2016. Healing Magic and Evil Demons: Canonical Udug-Hul Incantations. Die Babylonisch-assyrische Medizin in Texten Und Untersuchungen, Vol. 8. Berlin, Germany: De Gruyter.
Reiner, Erica. 1958. Šurpu: A Collection of Sumerian and Akkadian Incantations. Archiv Für Orientforschung, Vol. 11. Beiheft. Graz, Austria: Im Selbstverlage des Herausgebers.
Reiner, Erica. 1995. Astral Magic in Babylonia. Transactions of the American Philosophical Society, Vol. 85, Part 4. Philadelphia, PA: The American Philosophical Society.
Scurlock, JoAnn. 2006. Magico-Medical Means of Treating Ghost-Induced Illnesses in Ancient Mesopotamia. Ancient Magic and Divination, Vol. 3. Leiden, Netherlands: BRILL.
Wassermann, Nathan & Zomer, Elyze. 2022. Akkadian Magic Literature. Old Babylonian and Old Assyrian Incantations: Corpus — Context — Praxis. Leipziger Altorientalische Studien, Band 12. Wiesbaden, Germany: Harrassowitz Verlag.
As suggested in my original reply, Babylonian magic, like Sumerian, doesn't really lend itself well to being overlaid onto traditional Western Ceremonial systems. If you really want to make use of it, it's best to study it independently of other traditions so that you can appreciate all of its own unique nuances.