You can try having the same hostname resolve on the lan to the local IP and only connect using that first part of the addess, having dns suffixes do the heavy lifting

Yeah you should be able to set a custom dns for your whole tailnet I think, going by memory. I know I’ve had some issues but just try it, it’s quite fast to set up :)

Yes that’s what you can use the tailscale subnet routing for - one some machine on the lan, that joins the tailnet, you configure it up to route to local subnets