I knew they provided some "perks" to incentivize using their own router, like free support and compatibility with other junk they push to customers, but actively forcing users should be forbidden.

Do you also have to pay to "rent" the device?

Yeah I have AT&T and had to set up IP passthrough on their router/gateway box. Basically it makes it so the ISP provided router acts as if it isn't there and my router gets to do whatever it wants.

Just stick a router behind that router?

If it's fiber, you don't need the modem. You'll still need it once every few months.

Things you'll need:

1. your own router
2. cheap 4 port switch (1gig pref)

Setup: Connect gpon (the little fiber converter box they installed on the wall near modem) wan to any port on 4port switch. Then from switch to gpon port of modem (usually red or green port). Make sure modem fully syncs. Once this happens, you can move the cable from the modem to your own routers wan port. Done! Allow router a few moments to sync as well.

Now, every once in a while they'll send a line refresh signal that will break this, or if a power outage occurs. In such case, you'll just plug back in their modem, move cable back to gpon port of modem, wait for sync. Move cable back to router.

Edit: (after thought) put all this equipment on a battery backup and you'll still have Internet during short power outages.

What you could do, is set your phone with a temporary static IP (like, manually set on the device if the router doesn't have static leases). Then port forward to your phone as you would for the server. Then, set the phone back to DHCP, and set the server as the same static IP you used.

Assuming the router isn't smart enough to try to follow your phone's IP, you'll effectively have forwarded for the server.

It may also do it based on DHCP provided names, rather than WiFi names. In that case, you should make sure the server uses DHCP and advertises a valid name. If it's already got a static IP, that would explain why it doesn't show up on the UI.