I worked in a gym while in college and an issue similar to this came up. This was years ago, so the "awareness" wasn't the same as it is now.
There was a pre-op male to female transexual who frequented the gym for years. Everyone knew this person and what was going on. There was never a problem. At first this person used the men's locker room. They mostly came ready to workout and didn't change, it was more for just using the locker to store valuables. As this person came closer to their operation and was becoming more and more female, they became more uncomfortable in the men's locker room. This person met with the gym managers, and then with the managers and all the staff.
The solution was simple. This person would start using the women's locker room. They never got changed anyways. As far as actually using the restroom, there was a private unisex "family restroom" that they had always used.
During the whole time I knew this person, regardless of which locker room they used, they were very courteous, discreet, and polite. We never had one complaint from someone who felt uncomfortable.
During the meeting we had, this person was mostly worried about making other people feel uncomfortable and what could be done to alleviate that. They hadn't even originally asked to use the women's locker room, all they wanted was a place to store their valuables without having to go into the men's locker room. The management decided that it was ridiculous that they wouldn't have access to a locker room so they asked if this person would be comfortable in the women's locker room. They tried it out for a few days, all went well, so we all had a meeting together and said this is how it's going to be. If someone has a problem with it, they're welcome to change in the family restroom and store their stuff somewhere else.
This was probably a different situation because it was a small gym in a small community and everyone knew and liked this person anyways. It also probably helped that they were always super polite and considerate.
That said, I really don't see how this is an issue as long as everyone is polite, courteous, and follows basic locker room etiquette. It's entirely possible that the transgendered person at Planet Fitness was being inappropriate and walking around naked and gawking at people, but chances are they probably weren't. They probably, got in, did their thing, and got out without looking at anyone and not showing anything. This really shouldn't be an issue.