With this amount of traffic you have to load-balance, so you can't just set large TTL.

Look, you could just set the new subdomain as a cname with a large TTL and then handle that a-record via a different provider if you really need to. You could also choose to delegate the whole subdomain. Also, load-balancing via multiple a-records does not at all depend on the TTL. DNS is really the easiest part to handle here. Internal policy might prevent delegation, but that’s not a technical problem.

How can I handle a subdomain with a different provider while handling the whole zone myself? I'm not an expert in DNS, so really curious.

Set an NS record for the subdomain on the parent domain.

Is it cached the same as A records?

It’s cached according to the TTL you assign it, though it would be common to give it a high TTL.

But load balancing also applies to their new domain as well. Why would you need a separate hostname?