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I think you can get a free subdomain and dynamic DNS service at https://desec.io, with this you should be able to keep the domain updated with your IP and point it at your home server. But you need to have a public IP from your ISP and not to be behind NAT.
Anither option is to use a Tailscale Funnel. You will have to use a .ts.net subdomain with them and they terminate TLS and re-encrypt for you. On the other hand it’s completely free, you get NAT traversal, an encrypted tunnel, and you don’t have to maintain the IP even if it’s dynamic.