how about the gym mats over the concrete floor?

you can buy those and put them together like puzzle pieces. easy to take up, take outside and hose off to clean.

I suppose you could lay down tile or laminate and use these over it

I was going to suggest a large gym mat with good padding to protect your joints and your weights .

https://www.homedepot.com/b/Flooring-Exercise-Gym-Flooring/N-5yc1vZb97i

What's the gym being used for? Free weights? Resistance machine? Pilates machine? Yoga? Dance? Basketball? Hockey (floor, ice, field?)?

Google 'horse stall mats'

Durable, solid, hold a ton of weight, needs nothing to install them, removable when you leave.

TSC Tractor Supply has a 4'x6' 3/4" mat for $45

https://www.tractorsupply.com/tsc/product/4-ft-x-6-ft-x-3-4-in-thick-rubber-stall-mat

And just for fun, rubber mats give off a heavy odour. Especially the 'horse mats'. For those of us who are allergic to latex, the odour can cause breathing issues - ambulances may or may not be required. Nope. Not kidding.

If anyone in your home has asthma, COPD, emphysema or an allergy to latex/rubber, you will want to rethink the rubberized flooring.

I was worried about this. It is a small room but there are no windows or a way to “air it out” - we would use this room for light weights, peloton, HIIT, yoga. I settled on the rubber tiles - but I hope to let them air out before installation and I purchased a special cleaner.

3/8" thick Amorim (or other) roll rubber. It does have a slight smell initially but it dissipates quickly. I've done this in high end houses and have some in my own garage.

It takes a HIGH END rubberized product (as mentioned by Johnson Flooring above...Amorim = highend cork/rubber products) to get a non-odourizing product.

Airing out a low-grade/smelly rubber = forever. I'm not kidding. Some of these products STINK for months and months and months. Some people have had to throw them out to get rid of the smell.

When the rubber arrives, go ahead and open up a package and leave it sitting beside your sofa (or wheresoever you like to sit during the evenings) and see how much it affects you. If you get a headache inside of a few hours (my headache/migraine starts within 20 minutes) or you find your lungs feel like they are 'burning' or like they are feeling 'tight' then go ahead and send them straight back to the company you purchased them from.

Of course you will 'own' the package you opened...to be 'sure' of your decision leave the package open for a few weeks in your living room. After 2 weeks, if the odour is still there then it will be one of the one's that take months and months and months to dissipate

Personally I would have purchased a cork floating floor for $3/sf and clicked it together. The type of use in your little gym is perfect for cork flooring.