Congrats but there's no such thing as a factory 360 in a CJ of any kind. The 304 was the only V8 Jeep ever offered in the CJ series. Externally they are identical, so you can't tell by looking but the displacement is embossed on the side of the block under the motor mount.
Actually I'm not sure the verdict is in. As I said, there are plenty of resources that give help on decoding the VIN number on Jeeps and many of them show that there is such a thing as a Jeep with a VIN that specifies the 360. I'm about to get a flashlight and go check mine.
This page, (https://blog.jeep.com/heritage/a-his...-cj-6-1956-75/
"This quest for power took another step forward in 1973, when parent company AMC shoved a 304-cubic-inch V8 (or the optional 360-cubic-inch) under the hoods of all CJ models."
Also, look at this page and examine the details for VIN Digits 7, and 8-13
Such as, An "N" in Digit 7 on 1975-1977 CJ5 indicates a 175HP 360. For model years 1978-1980, it indicates a 360 with a 2bbl carb.
1975 - '77: V8, 360 CID, 175 HP
1978 - '80: V8, 360 CID, 2 Bbl.
If it's a "P", then it is...
1975 - '77: V8, 360 CID, 195 HP
1978: V8, 360 CID, 4 Bbl.
So clearly AMC offered a factory 360 at the very least as a special order option.
I just checked my VIN, and the 8th character is an "N", which indicates that mine was indeed a jeep that shipped from the factory with a 360 / 3-speed and a two-barrel carb.
Here is a screen-shot of my VIN check, with my VIN (the last five digits are redacted) showing that it came with a factory 360.
You can run a vin check on yours for free here...
So... I don't know, but I feel pretty strongly that my jeep came from the factory with a 360.