I don't see why your HTML wouldn't work in CBs emails. CB now just uses Joomla's mailer so both should be sending emails the same way unless AcyMailing has some additional processing. What about the emails is not displaying properly? It's likely going to be easier to try and fix the HTML than to completely replace the emails in CB.

Looks like there's some broken HTML somewhere that AcyMailing is probably fixing for you or you're using features that only exist in AcyMailing in your emails. I can't really help you with this. You'd need to review and fix your HTML yourself. CB doesn't apply any processing to those emails beyond sending them through substitutions which don't do anything if there isn't a substitution in the contents.