Internal DNS for SaaS Tool?

We're using a SaaS tool where we are able to have it accessible via our domain name.

support.mydomain.com goes to me.tooldomain.com via  CNAME record. The DNS entry is identical on our external DNS (Cloudflare) and internal (Server 2019).

This works perfectly fine outside of our LAN.

However, on our LAN if users try going to support.mydomain.com they get a "NET::ERR_CERT_COMMON_NAME_INVALID" in whatever browser they use.

Specifically it's saying that the server could not prove it is support.mydomain.com and that it's security certificate is from *.tooldomain.com.

This happens with all major web browsers. I just don't understand why we're only having problems on the LAN side, and the DNS records are identical.

Any help would be greatly appreciated.