If it's a public pool and you're a resident in the municipality, they CAN'T kick you out.
You pay taxes that support the facility.
The most they can do is insist you wear a rash guard in the pool.
Bingo. Legally they cannot kick you out - since is a public facility that would be discrimination.
They can require you to cover up, though.
That said, some places are more chill and some less and with all things in life you need to pick your battles.
If the other places don't work out and you're still hard up for a gym it might be a good idea to try and meet with the manager there and see if there's a way to make things work (i.e. rashguard in the pool, longsleeve T or Underarmor in the workout room, etc).
Took a quick look on the Gold's site but it seems like the nearest ones are in the Kansai region, so not really an option (unless I missed something).
For setting up lessons, there are sites out there like this -
https://www.getstudents.net/
But to be honest it was over 10 years ago when I was doing lessons and it was in completely different regions of the country to you so I really can't provide any relevant references.
I do remember that way, way back in the stone age (15 years ago?) when I was living in a smaller town there was an international center there that had a bulletin board (literally a bulletin board - pieces of paper stuck to a corkboard) for people looking for teachers or teachers looking for students. I have a feeling almost all the opportunities in a big city would be through online services but you'd be surprised how slow things can be to change in the more regional areas of Japan.
This is the site for the international center in Okayama -
http://www.opief.or.jp/
Might want to swing by there if you have a free afternoon and see if they have anything like that!