An internal rewrite using a RewriteRule could solve the URL problem. Continue with extensionless URL requests on the web and rewrite the request so that a .html file serves the content without revealing the name of the file. It's a couple of lines of code at the most.
You do not want a redirect using Redirect or RedirectMatch. You do not want a redirect using a RewriteRule. You do want a rewrite using a RewriteRule.
However, dumping the CMS sounds crazy. You lose the ability for timely updates to the site, the ability to make site-wide changes in one edit and a whole host of other useful features.
If you're migrating off the current host, you should be migrating to a new CMS. If that's not urgent, take the time to learn PHP and mySQL and do the job properly.