If they did show the Enterprise, the Stargazer bridge is similar enough to the E-E bridge that they could get away with redressing it slightly and it would be the E bridge with 20 years worth of upgrades. Bit like how it was used for the brief glimpse of the Excelsior bridge we saw in S02E01.
As it happens, Doug Drexler posted a behind-the-scenes photo from season 3 last week of the Stargazer bridge set, which he quickly yanked down and redacted, since there's a spoiler in one of the background consoles. It's just blurred, so you can tell that it's the dorsal view of some sort of long-ish standard-shaped starship. Could be a Sovereign. Or an Odyssey. Or any of the various Excelsior and Excelsior-eqsue designs we've seen. Maybe a Vesta. Or something new. But the upshot is that it's playing the bridge of a ship that's apparently not the Stargazer.
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