[PZ]

You would need to match the same gear ratio unless you change the rear differential. The FAS had a 4.11 ratio, but I don't have the ratio for the FEP.
Different years had different electrical component locations which complicate swapping between the different trans model codes. I found swapping to a manual trans to be the easiest/cheaper solution rather than having the trans rebuilt.
You would need the 6sp manual from either a B5 S4 or a C5 A6 2.7T (not Allroad).

 

[PZ]

I've had a couple where the TC pulled the bushing out of the pump. If you have the tools and clean work space, a rebuilt is a better idea than a mismatched replacement.
There is a temp sensor that can either be in the wiring harness or inside the trans depending on the model. There is also a speed sensor of some type (input or shaft?) that is different between models. I'm not an expert on that and never really got into the internals of anything above a TH400 about 40 years ago.

I have the PDF of the 01V trans, hopefully, the attachment works.

 

[PZ]

Was the TC pushed back behind the opening for the engine speed sensor? I suspect it is not engaged completely. It should be about 1" away from the flywheel when the trans is installed.

 

[PZ]

The only other issue could see causing this would be a problem with the pickup tube on the filter.
This is how far the TC needs to be back, completely behind the sensor opening. I felt 3 distinct clicks as each part engaged.

 

[PZ]

Did you have to pull the TC forward to get the bolts in? If the TC was already touching the flywheel, it might not have engaged.

 

[PZ]

I use a prybar through the starter opening. It sounds like it should be in properly. Is there any fluid running through the trans cooler lines ? If not, I would start with confirming the filter seal is in place and tight. Is it possible that the replacement trans has a bad front pump?

 

[PZ]

I haven't pulled apart a trans in 40 years. Could it be the TC was not fully seated and put lateral pressure on the pump?