Only one problem with that. There was only one Excelsior class ship at the time of Star Trek 4.
Actually, we don't know that at all. All we know is that Excelsior
was the first. A second Exclesior-class ship could have been waiting in the wings in the final stages of construction at the time of Star Trek IV, and that ship became the new Yorktown
And even it would be refitted somewhat between it's going from NX-2000 to NCC-2000.
As far as we know, Excelsior
's registry was changed shortly after Star Trek IV after a second Excelsior-class ship entered service.
Starfleet doesn't have to worry about PR in my opinion. Because if they were going to really do that then it would be more of an honor of having a brand new ship named Enterprise than simply, as you called it, slapping the most famous name in the fleet on a hand me down.
I disagree. Designating another Constitution-class ship as the Enterprise
and putting Kirk in command was a nice PR move (as well as a means of honoring the ship). The true origins of the ship as an earlier vessel would be irrelevant to the public.