So I was wondering, does Danse become unavailable as a companion if you join the Institute?
Actually, this isn't entirely true. You can follow the BOS story missions up to BUT NOT INCLUDING the mission "Tactical Thinking" without hurting your reputation with any other faction. If you follow their story line that far you will complete the mission "Blind Betrayal" which will remove Danse from the Prydwen (if you haven't already moved him) and will remove the threat of him being eliminated with the rest of the brotherhood if you decide to side with the institute. This mission also rewards you with a set of pretty good power armor so I find it's worth it to follow their story line through Blind Betrayal just to get it.