I always envisaged that by post-Nemesis era Soveriegns would have replaced Excelsiors, Novas replaced Oberth and maybe only the Constellation-class of the TOS Movie era would be around.
Some of the semi-canon ships are interesting but others are frankly repulsive.
Personally, I think Intrepid class is a great choice. I think its a class of ship that would have become dated and so would go out of production quickly but would likely still be in service.
I wouldn't think many Intrepid-class ships would have been built. The retractable nacelles I thought were Starfleets response to the subspace damage caused by warp drive and I would assume that since later classes e.g. Sovereign, Akira etc... were built with fixed nacelles that warp drives had been improved.
Also, an Intrepid class ship that stayed in the Alpha Quadrant (unlike Voyager) would probably have quite a history. It could have been through the Dominion War, for example, so it would have history.
Only other suggestion would be to look at the other classes of ships that featured in the Battle of Sector 001 in First Contact. Could go with a Sabre-class, Norway-class or Steamrunner-class?