I have been on iCloud custom domain email for a while using Apple apps, and just tested it with Postbox (a 3rd party email app) on the Mac.
I had no problems receiving and replying to email addressed to both my icloud.com/me.com and custom domain email addresses. Even though they all came in through iCloud mail servers into a single in box, in replies, Postbox used the correct FROM address according to the TO address the original replied-to email was sent to.
When I installed Postbox, I gave it my iCloud email info and followed their directions for setting up the required app-specific password. Once email was working, I entered my custom domain email āIdentityā in Postbox preferences. That worked. One difference from Apple Mail is you donāt get the rapid push email from iCloudāPostbox has to poll the IMAP service. (Also I prefer table-style list layout, which I donāt see in Postbox)
While Apple canāt vouch for 3rd party app behavior, and neither can I, I canāt see a big problem, at least with this app.