They would probably still show up as "?", but there is a workaround (such as this) for this with OpenWrt on both the DHCP server and the dumb AP.
Yeah... sounds like it. Seems like any router would be better than the Starlink one. My knowledge of their device is minimal -- just a few things I've seen in youtube videos and such. But there are some good ones that describe using Starlink a an uplink to a more sophisticated network architecture (better routers, etc.).
Back to the hostnames thing... it's important to differentiate between "?"'s showing up in the wifi clients list vs the actual IPs/hostnames of those devices. Hostnames not showing up in the AP doesn't have any consequence except for the fact that you can't easily identify which devices are connected to the AP. But the actual IP addresses and the relationship with the hostnames doesn't change as a function of the AP association, as it is all handled by the DHCP and DNS server.
So, when I consider your situation, your main router, regardless of the firmware on it, just needs to be able to assign and DHCP reservations and associate the hostname within the DNS server. As long as that is working, your network addressing (by IP and/or hostname) will work regardless of the AP that happens to be used (this is assuming the use of dumb AP configuration in the other buildings and a better router in the main house).