I wouldn’t call it a “fine-dining atmosphere” if they let you bring your own alcohol, wine or not.
If the restaurant is “cheap” enough to have to allow people to bring their own wine (they either don’t have a full license, have a terrible wine list or need to appeal to skint people), I can’t see them refusing to allow beer…
Maybe it’s different where you live though, do many places allow you to bring your own?
In the UK, it is really only curry houses and little cheap restaurants that allow it, and even then it is quite rare, and definitely viewed as being “cheap”, whether you bring beer or wine. Wine more so, in fact, it being generally cheaper.