This will be tough... Normally, you would sacrifice range if you want a good wheel/tire set up. Most likely, you will end up with a wider wheel/tire combo from aero stock wheels. You can always stick with stock sizing but where's the fun in that?
19 vs 20 will be prefernce but you are correct. Most likely, 19s will ride better and will have a bit more rubber to proctect you wheels. Not saying 20 isnt going to be confortable but 19s is likely to deliver a more comfortable ride.
If you want the titan t5 design, check out SSR GTX01 in black. Very similar and priced around $2K. Plus tires, you will be right around your $3K budget... BUT you gotta think about your lugnuts AND TPMS sensor that will push you out of your budget (TPMS new is about $300 for a set if you do not want to move your existing TPMS sensor from 1 wheel to the other).
Tires, Falken FK510 (Summer tire) or Conti DSW 06 (All Season) for daily driiving.
Hit up
@P3D-R He will help you out.