Most bad exhaust smells come from, either the cats (usually rotten egg smell) and or the fuel.

Transmission fluid is known to smell of cat piss when warm/hot.

Are you certain that you do not have a transmission leak onto your exhaust. The exhaust would very quickly burn it off when the car is hot. At start up it will cook nicely for a few minutes before it gets hot enough to vaporize it.

I would be tempted to get under the car while it is cold and run your hand along the top of the exhaust to see if there is any trans fluid pooling there. Also have a general look to see if there is any evidence of dripping/pooling near the exhaust tunnel, lying on heat shields etc.

Please do report back.