a question. I No it should work this way: I have a WiFi router with my SQ5, and I can connect to it, which allows me to operate it. However, once I’m connected, I only have a WiFi connection to a network that doesn’t have internet access. Would it be simpler if I could just use my mobile device, and could the app handle this as well? Or do I need a router with a SIM card? Is there a different way to set this up?
Iphone 15 
I’m guessing most of us set our mixer WIFI up on it’s own stand alone WIFI router without
internet access on purpose to keep traffic on that network to a minimum.
At a couple installs the mixer has been put on the buildings network but on a dedicated hidden sub net.
This is pretty typical. Most people prefer their tablet only control the console (in this case, SQ) and not be getting notifications, etc., for other apps via the Internet. Your phone should, however, pop up a message saying there is no Internet on the connected wifi and ask if you want to use your cellular data. This is what I usually see, at least.
Otherwise you do it like Mike C said in his reply.