To clarify, This is what I have:
https://www.ebay.com/itm/155065766086?nma=true&si=MmbRQ601PC5CgEY138kHApcbDZY%253D&orig_cvip=true&nordt=true&rt=nc&_trksid=p2047675.l2557
I bought it from that seller off eBay. It's currently in a 2004 2.7L car that runs, but shows an error for ABS coding. If nothing else, I figure that it could help with your diagnosis and only cost shipping to try it out.
To test your car, I would send the DME, Central locking module (immobilizer box), and the key fob - which houses the transponder for the immobilizer. It is from a Tiptronic car, but should be able to run the engine in your car - is the DME in your car the original Tiptronic?
I also have a DME and immobilizer box that originally came with the '04 2.7L car, but no key fob to go with it.