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Theoretically, the .45 ACP and Auto Rim are identical except for the rim; however, the .45 Auto Rim cartridge is rated at a lower max pressure than the .45 ACP. This is most likely in deference to the old M1917 revolvers for which the cartridge was originally designed. For a modern S&W 625, most folks use .45 ACP data for the Auto Rim.
The big question is what does the maker of the .45ACP outlaw cylinder recommend? Some of the cartridge conversion units for BP C&B revolvers specify BP only. Without other restrictions, I'd use .45 ACP data that doesn't exceed the Auto Rim's 15,000 CUP max. Since I have many home cast 200 gr SWC's, I'd probably start with them over 4 gr of Trail Boss.
Hodgdon(IMR) shows .45 ACP Trail Boss loads:
180 gr LFP 3.5 gr 664 ft/s 7,100 CUP -to- 5.0 852 ft/s 13,700 CUP
200 gr Cast LSWC 3.5 gr 652 ft/s 9,200 CUP -to- 5.5 816 ft/s 16,100 CUP
230 gr LRN 3.5 gr 658 ft/s 11,200 CUP -to- 4.5 761 ft/s 15,100 CUP
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