Audi will only fix the problem if you have scrupulous records of oil changes according to their schedule.

If it is a sludge problem, let the other guy take care of it.

To drop the pan probably costs 5 hours labour. Not a cheap job.

 

Update:

I'm still not sure of the exact problem that was causing my startup issue. My car had a sludge issue + a bad oil pump, so that was taken care of. They also cleaned out the throttle body. I no longer have the problem, though I'm sorry to say I don't have an exact solution for you.