One approach is to put the wildcard certificate on a proxy. Clients connect to the proxy, and see a valid TLS certificate. The connection from the proxy can use whatever you'd like, including a self-signed certificate that you pin. You have the impersonation problem if the proxy is compromised, but that's a smaller attack surface than "printer", etc.

To be frank that's one of those things where it depends on how much you care about what gets compromised. For a purely internal homelab, the main use is to get software working; the HTTPS is only needed because modern browsers get upset about a compromised secure context if you don't, which leads to them disabling all sorts of random APIs. The actual security component is pretty much secondary for the usecase where you're dealing with devices that exist solely on an intranet - you're already gating off access by different means.

Personally, the way I'd solve the problem in the situation where it'd be a problem is to not put auth on the same domain; auth would go to auth.secondexample.com. That's generally speaking the better choice to begin with due to browser weirdness regarding domain isolation (which is also why you often see software with CDNs use completely different domain names; reddit images being hosted on redd.it for example but also YouTube thumbnails being on ytimg. If they didn't, a maliciously formed file could read out browser storage for different subdomains).

> somebody compromising printer.lan.example.com can impersonate auth.lan.example.com

Nope: one can setup ACLs so that the script that can update the records for printer.lan… cannot also do foo.lan… or any *.lan….