Single Site on multiple domains with CNAME?

I know what I want to do, but I don’t know how to implement it.

I’d like to have my site accessible on a different URL to what I host it on.
For example, let’s say I own:
[1] example.com
[2] eg1.eg.net

Requirement:
I don’t want to ‘transfer’ my main domain to my web host as I want to keep using my registrar’s mail service.
So, I’d like to host my site on a second domain, but have visitors access it on the first only.
My visitors must not know the second domain exists, and my web host must not know the first domain exists.

Plan:
Redirect example.com to https://www.example.com
Map https://www.example.com to eg1.eg.net using CNAME

Issues (what I don’t know how to do):
I hear that WordPress doesn’t like running on multiple domains because of the way it handles permalinks – I need all links, relative and permanent, to work on https://www.example.com

It would be great if I could redirect all attempts to browse directly to eg1.eg.net to https://www.example.com or something else entirely without losing functionality on https://www.example.com.

Search engines must find my site only on https://www.example.com
I have a feeling that achieving the above redirection may solve this one too.

——
I’m using a single site, but may want to change to multi-site after a few months.