Could just be "Enterprise Class" was just the version of the simulator, since the Enterprise was the Academy ship at the time, they'd use an Enterprise bridge as a template for the simulator since it would be the ship they'd first serve on.
It's possible. But even in your own sentence you say they'd use an "Enterprise bridge as a template" instead of "Constitution bridge". The simplest explanation is still that the simulator was for an "Enterprise Class" ship, IMHO.
Back to the schematic in TUC. All these years I never once looked at the "pen" Scotty is using. It looks like a telescopic pointer stick repurposed as a futuristic pen prop. Or perhaps it was a pointer and he was going to use it to point in a presentation to direct search parties to look as jackoverfull suggests?