Hi Linus,
I can only imagine how challenging that would be...
I work in the fitness industry (with seniors) and know that many of the newer facilities (especially YMCAs) have "family" or "assisted" changing rooms...basically a single small room with lockers in a hallway area outside the room that can be used to secure your belongings.
It can be a bit of a hassle to wait for the space if you're there at a time when lots of families are using them, but it eliminates the "where do I go" problem. I used them alot when my son was 8 or 9. I just wasn't comfortable sending him into the men's locker room alone...and he was too old to be in the women's...so that was our best choice.
Best of luck with your fitness goals!