Show of Hands

If she over-serves her, probably, at least in some states, but not just because the woman showed up pregnant. How would the bartender even know the woman was pregnant unless she asked, and why would she? Most of us learned at a fairly young age how risky it can be to ask a big girl if she is pregnant - because if she isn't you don't know for sure you can outrun her, and she is likely to react badly to the question.

No - they aren't belly inspectors.

Isn't that how people become bartenders?

Maybe the same as someone helping another do drugs? It'd also depend on how clearly pregnant though. If there's no reasonable expectation that she's pregnant the bartender should be free and clear (like obese women for example).