First, your local gym staff definitely can't help you with getting the app to work. However, they are probably nice people and incentivized to get as many customer as possible so if you tell them something along the line of "I'm not good with technology and I usually don't like to carry a phone around" at worst they will look at you a bit weird but they will help you find a way around the app if there is one.
If there is no way around, you are perfectly entitled to contact their technical support (if you can find contact info on their website) and inform them that you are unable to take advantage of the subscription you paid for because their app is incompatible with your phone by lack of GPlay. Sometimes, it's enough to put them in front of the problem to make them realize that their technical choice can impact their business. They will probably ask you what phone you have but if telling them that you have a pixel without GPlay will result in them brushing you off as "it's your fault", you can also bullshit them with something along the line of "I worked/lived in China until recently and have a huawei phone". If this works, it will take at least a few weeks if not months for the fix to be released but it definitely can work.
Finally, the simplest way remain to just install GPlay and the app in an isolated profile. It may work without signing in to GPlay and even if it doesn't, a gmail account can be entirely anonymous. Sadly, the Equinox app has your identity and may leak it to GMS depending on how they do things.