Is the phone is Low Power mode? Does the browser have Private Mode turned on?

I don't think so. And if it connects to other wifi access (outside the building) it doesn't turn off. Only in our establishment.

I don't think it's really possible to help you here. I can tell you that my hotspots work with iOS correctly. and I can tell you that if iOS is in low power mode, or the battery is getting low, it will release the dhcp lease and turn off the radio when it goes to sleep. But without actually interacting with the hotspot and the devices in question, I don't know how much anyone could help.

Personally, I don't use public WiFi on my phone, so I don't know what it does when it locks. I would suspect that it should disconnect, as being connected to an open network is never a good idea.