If you are thinking about going into town for dinner or a bit of evening sightseeing, then definitely stay at a hotel in Sultanahmet or Taksim.
If you're after convenience as this is a purely MR trip, then I'd stick with the airport hotel.
The "E" gate is accessible for all *A Gold card holders, not just TK Elites. Like others have said, I have used this gate even while flying non-Star airlines by showing my gold card. On a few occasions when I was dressed more formally they even waved me through without checking any credentials.