There's a masjid which offers a gym inside their masjid. They are upon bid'ah however there are no females or room for fitnah in terms of what we see in traditional gyms. Is it permissible to go there?

I don't believe you mean inside the masjid itself. I think you mean maybe next to the masjid or something like that; this is allowed inshallah. But inside the masjid—like inside, inside, where you pray and so on—of course not.