Want to use the Omnia router as a very-well-endowed AP

Since you are using Lan0, I assume you run Turris OS 4 (or higher).

The basic configuration should be to disable DHCP and create a bridge containing all lan and wlan ports. However, there are issues in Turris OS 4 where the internal switch seems to not update properly when a device moves around in the network, i.e. moves from one AP to another. That will cause DHCP to fail, so you might be running into that. See OMNIA: Vlan on DSA port breaks arp responses (TOS 4.0.5)

If that is the issue you are having you should see that DHCP fails if the device has been connected in an other way recently. However it should start working again once the device has been of the netwerk for ~20 minutes. For wifi devices, in Android 9 and higher you can enable mac address randomization which will avoid this issue. I think the same is true for windows 10, but I haven’t tested that.