What annoys me even more, regarding to my initial posting. I had never any issues with any router/adsl- or cable modem. I used 5 different ones the last 2 decades with WEP or no encryption without any problems. Just now my new phone gives me just "half internet access" :-/
I've been thinking about your issue off and on as I'm still troubleshooting my intermittent Airport issues (random disconnects, sometimes no internet access but LAN access is fine, etc.). I wanted to break your situation down a bit further and maybe get some clarification if you're up for it.
I assume you're on OS 9.2, but which version of Airport are you running? I've just updated to 2.0.4 and turned off "Interference Robustness" myself and was surfin' the net last night (all 3 sites I can access) without issue. This morning, however, it gives me like 5 minutes of good internet connectivity before it stops connecting (but I can still access my LAN server just fine), so it seems my pancea wasn't. I'm now having to turn off my iBook's Airport and back on to get a few more minutes of internetting.
Since you're hotspotting from your KaiOS phone, I assume other devices can connect to that hotspot without the issues seen on your Ti? It sounds like there's not many user configuration options on the KaiOS side, and you said you turned off wifi encryption so that shouldn't be the issue.
With my original Airport Base Station set into bridge mode with DHCP/NAT disabled, my iBook's Airport control panel also displays the "Getting status.../Status not available" loop every few seconds. I assume it's because DHCP/NAT is disabled and Apple didn't care to include status message handling for bridge mode.
For your TCP/IP settings, I assume you're set to DHCP, does the IP assignment look correct (IP has been assigned, not on the localhost 169.254.x.x subnet, etc.)? Since you said you can surf with Classilla and find a file to download, I assume it's set correctly, so it could be some protocol on the cell phone carrier's side that cuts your session short.
When your download stops transmitting, are you also unable to surf the net at that point? And is the Ti still connected to the KaiOS's hotspot?
If you don't download anything, how long can your surf the net? And I assume your KaiOS phone is positioned near your Ti computer for optimal WiFing, but are there other bluetooth or wifi devices nearby that could be contributing interference? And is there a way for your KaiOS to change which WiFi channel it's operating on?
Edit: Also, and I keep forgetting, if you have a dead battery your system clock might not be set correctly. This prevents me from connecting to the internet, either because of my router or ISP or mage that guards the portals. I won't get any messaging that my system's clock is set incorrectly, with the web sites Classilla will just time out on the "Connecting to <website>..." and stop trying. Even trying to sync my system clock via the Data & Time control panel doesn't work, which makes me think it might be an issue with my router (maybe even the Airport Base Station) rejecting packets because the timestamp is off by decades.