I’d recommend getting your own domain name and using a full DNS service that has an API. deSEC.io is free and well supported by common tools like ddclient, but you can also use their API directly with one curl command.

Or you can use your registrar’s DNS servers if they have an API, that would also be free most likely since it would be included in the service.

gandi.net has excellent DNS services included with domains but they’ve recently pumped up their renewal prices and now you have to dig a bit through their offer to find something below $20. There’s .fr, .gdn, .gr, .in and .uk domains (including .me.uk).

I’d recommend an EU country like .fr or .gr if you’re an EU citizen because then their registry data falls under GDPR, plus most EU ccTLD registries are private by default. Otherwise .in also has good whois privacy and is the cheapest.