I looked at the footage on Hulu and Kirk views the ship from the port side, not from the bow. This leads me to believe that either 1) he could see the bridge crew through the transparent dome; or 2) the shot of Kirk on the main viewer was inserted later in some effort to provide context.
The dome isn't transparent, though. In the second pilot, when they enter the bridge, you can see a fair amount of the bridge's domed ceiling, and it's opaque. The opening shot of "The Cage" implied a transparent dome, but if that was intended literally and not metaphorically (as a "cutaway view"), it's an idea that wasn't kept beyond the first pilot.
I've always figured that the scripted intent was that Kirk was looking through the saucer windows, but they didn't have any footage of interior windows, so they just used a stock viewscreen FX plate as a stand-in. It was silly, since anyone who'd watched the show before and paid attention would know that you couldn't see
in through the viewscreen, but it was the best they could manage to convey the idea.
Plus, of course, it was the third season, so attention to logic and consistency wasn't always a thing. Heck, the whole "Flint shrinks the
Enterprise and by the way also freezes it in time" plot point was totally gratuitous and nonsensical anyway; the viewscreen silliness is one of its lesser problems. (Between TOS and
Space: 1999, it's clear that a hallmark of Fred Freiberger's approach to science fiction was giving the antagonists whatever random, extreme, and nonsensical superpowers were convenient to the plot at a given moment.)