2001 4.0L SOHC Swap
I am not sure what the PCM number you noted is exactly. BAE4? 2001MY PCMs were made by Motorola in their plant in Seguin, Texas. The base part number for all such PCMs is 12A650. For 2001MY the prefex/suffix should be either YU7A/APA. 1U7A/CCA, 1L2F/AAD, or a 1L5A/BDB,BFB,or BBB.
All 4.0L SOHC PCMs since 1997 through 2001 were based on the Ford ML2 PCM design, so pin-outs should be very similar for all versions, but calibrations and features will vary by year and application.
You need to also be careful to match the injectors to the correct PCM as Ford produced both normal gasoline versions and Flex-fuel versions of the 4.0L SOHC in 2001. Gas versions have blue injectors (19#) and Flex Fuel versions have red injectors (24#) Check which type of injectors your engine has before selecting a PCM.
If the PCM has software PATS, then yes you can have it re-flashed to delete the PATS functions. Best to check with one of the module re-programing companies to be certain.
The above confusion is why I suggested that you get a complete engine/trans/wiring harness/PCM set from a salvage yard or by buying a complete salvage vehicle. Once the parts are separated, it is difficult to avoid mis-matched parts.