IPhone no connection available for non internet Wi-Fi setting?

I have a few devices that connect to the phone via Wi-Fi given off by the device. It will connect to the network and then pop up with no connection available and disconnect. I assume since it is not an actual internet connection my phone is not allowing it to stay on the network. Is there a way to change this setting or bypass the problem?

Have you set up the phone's Wi-Fi with static address settings? If so, then it might be that Apple don't allow the hotspot from the phone to work without an Internet connection and the static addressing is not permitting it to reach the Internet.

Another alternative is that your phone has an incompatibility issue with your router. Such issues appear to affect between 2% and 5% of router/device combinations based on my experiences.

You need to ascertain whether it is a Wi-Fi issue (between the phone and the router) or an Internet issue where the phone can't establish an Internet connection.