Everything I mentioned works for LAN services as long as you have a domain name. You shouldn’t even need to point the domain name to any IP addresses to get it working. As long as you use a domain registrar that respects your privacy appropriately, you should be able to set things up with a good amount of privacy.
Yes, you can do wildcard certificates through Let’s Encrypt. If you use one of the reverse proxies I mentioned, the reverse proxy will create the wildcard certificates and maintain them for you. However, you will likely need to use a DNS challenge. Doing so isn’t necessarily difficult. You will likely need to generate an API key or something similar at the domain registrar or DNS service you’re using. The process will likely vary depending on what DNS service/company you are using.