When adding a new domain to the existing SSL SAN Certificate, how would it affect sites that were using the previous SAN Certificate?
My logical sense is telling me that it will be be able to serve https requests or throw a security warning indicating the certificate being invalid.
As long as a certificate isn't revoked (or doesn't expire), it will be valid for all the SAN names for which it was issued. If another certificate is issued for a set of hosts that overlaps the ones used in a previous certificates, both will be valid for their respective hosts.
Whether a CA chooses to invalidate a previous certificate it has issued within the same account for a similar set of hosts depends on its own policy, but in principle, you could easily get another certificate for the same hosts from a different CA anyway.
This being said, it seems that you're using a single certificate (and a the same private key) across multiple machines, for hosts that don't seem related (at least in the sense that they are served by distinct machines anyway). This is normally considered bad practice, since if one of those machine was compromised, you would have to change the certificate on all the machines.