Pretty permalinks incompatible with domain masking?

I want to be able to host many different website on a single host, but to point to them with distinct domains, masked so that no one is the wiser as to the physical location of the file.

So I want to have a blog hosted at the physical location

https://edgerstner.com/rants/

But I want people to see the blog as

https://silent-typewriter.com/

WP seems to thwart the domain masking that I have set for silent-typewriter.com, so to work around this, I’ve entered this instead as the location of the blog. That is, in the general settings of the dashboard I’ve entered the WordPress address (URL) as https://silent-typewriter.com/

This works fine for default permalinks. But if I set up pretty permalinks to have a month + postname structure, it all goes haywire.

Am I condemned to use ugly permalinks if I want to use domain masking. Or am I going about it the wrong way?

All help appreciated.