The bridge in TAS also has a second exit, where there is none in TOS.
Heck, there are gooseneck viewers in "The Cage" and "Where No Man..." that are all gone by the time of "Corbomite".
Canonically, the bridge has undergone modifications, some minor like the gooseneck and various station configuration changes, and some major like the second exit added when going from TOS to TAS. It's not a stretch to suppose that the automatic bridge defense system—a system not established in TOS—was completely new to TAS. Accordingly, it's entirely possible the dome was changed, too.