I would also suspect heat and " Although the tempeture gauge doesn't show over 120.." that's just too cold even for a raw-water cooled engine.
The way that it dies after warming up a little, and then runs fine once it is thoroughly heated up, makes me think "choke". That's the only thing that is changing with the warmup process, choke setting and a change in the carb from "cold start" to "main" operation.
Something like a bad coil or fuel supply would tend to simply fail as it heated up--and stay failed,
AFAIK.
I don't have any faith in any "rebuilt" carb, because rebuilding one properly takes time and attention, and most shops mess up something because they are trying to rush the job. I'd start by taking the carb and choke, breaking it down and rebuilding it literally "by the book", and making sure the choke parts were working correctly. If there's a vacuum diaphragm or something similar--check that for leaks and proper operation. Could also be the carb is not seated properly, and the changing heat is opening up a tiny vacuum leak between the carb and engine, then shifting to close it again, so don't assume anything, check that gasket & fit for vacuum leaks as well.
Be glad it is an A4, working on fuel injection systems can be much much harder.