Any suggestions
I don't know where you live, but if its in the US, I think its a tough road. I think I would take the money you would spend on gym membership and buy equipment for your house. Then you can workout however you please. That's what I do. Convincing a gym owner to open for nude use seems like it will create many problems for the owner.
I've often thought about this, particularly since the Y used to have naked swimming (back in their single sex days).
I think the best way is to demonstrate that it could lead to increased business. I've thought approaching our gym about seeing if they'd let me use their pool, either late at night or during other off hours. I suspect that if they'll let me do it at all, it would be for extra bucks. If that's the case, you'll want to split the cost with others.
If, however, you can persuade them that "clothing optional' hours could be good for business, then you can work with them to find more customers to make it an economic success.
good luck
My nude yoga instructor confirms that the couple of times he's had to change the venue, even temporarily, it has been pretty tough to find a place for our classes. He's managed it, but it takes knowing people who know people who might be OK with it.
For several years two nudist groups in this area cosponsored a naked health club night, where they rented out a small gym after hours one Saturday a month. You got to use the entire place naked, and I thought it was a fantastic social nudity event. I was able to attend a few times, and I'd almost forgotten how great it feels to swim naked. There was a volleyball game going in one court and I even got to work out on real gym equipment, something I'd always wanted to do. I don't know how it was arranged, but chances are the owner of the place was either a member or a friend of one. I suppose the clubs had to guarantee a certain amount to make it worth opening for it. They charged $10 each, and I counted around 40 to 80 people there. So maybe one way is to find a smaller place that would be able to make a few extra bucks by holding a private event like that.