From a high level, you need
1. GTM configured to handle a subdomain of your main domain (e.g global.thisdomain.com).
2. A delegation from thisdomain.com for global.thisdomain.com to your GTM servers
3. A configuration in GTM for service.global.thisdomain.com that resolves the IP addresses for you
4. Change the records in your public DNS for service.thisdomain.com to be CNAMEs for service.global.thisdomain.com
That's it. There are other ways (e.g. running GTM inline), but I prefer a subdomain. That way you can swap it in & out real easy just by changing the CNAME record in the main DNS...
The smarts (and the more involved config) is in the GTM config to resolve service.global.thisdomain.com of course...
H