Suggestions
Gym Profiles
Gym Profiles are profiles that the user can create which contains the name of the profile and information tying the gym they are in to that profile. This information can either be what equipment the gym has or what exercises can be performed in that gym; I personally prefer the first option.
Further more, there is a possibility to have a dropdown or toggle to select what equipment an exercise requires. The default exercises may have that information already set.
This would work even better with a larger directory of exercises, tying into a “collaborative exercise database”.