Are you talking about putting 15 monitors, one at each cout, or just one monitor at a central location for people to sign up? I wasn’t sure from your context.
Having to carry the tower or laptop, monitor, keyboard and mouse, plus the cabling is going to be a problem for most places here, so I don’t think it will be feasible for us. We also don’t have any storage space at the gym that we can use.
Your requirements are probably very different from ours. I know clubs run differently depending on size and purpose. That is another reason I only thought about the idea but haven’t acted on it. Developers need to make money for a living too. If we can’t make it generalized enough so we can sell a lot, we must be able to charge a lot for one or a few copies. It takes a lot of effort and time into making good software.
You could try to find existing queuing software. But, since they aren’t specifically made to solve this problem, it will never work the way you want. You will have to live with many limitations. As for open source, it is even less likely that you will find anything because the open source community consists of mostly reusable library code, not apps. We want to write code that a lot of us can use, not apps that only a few people find useful.