Can I set a CNAME to point to the CNAME of a separate domain [duplicate]

The standards discourage such a solution.

It probably works with most DNS software, but your clients might encounter some which is not willing to follow multiple CNAMEs in succession. In that case, those clients using such a DNS resolver will likely not be able to access your service using those domain names.

In RFC1034:

Domain names in RRs which point at another name should always point at the primary name and not the alias.

Of course, by the robustness principle, domain software should not fail when presented with CNAME chains or loops; CNAME chains should be followed and CNAME loops signalled as an error.