I'm posting this here to hopefully help others that run into this issue.
Good news: The new version of the Brave browser is based on Chromium, so you can install the Enpass browser extension there.
Bad news: There's currently an issue that keeps it from showing you some helpful information about configuring the Enpass Application to work with the browser.
Also, you have to uncheck the "Verify browsers" option, since Enpass doesn't seem to know about Brave yet. (Can it be updated to support that? Should I open a separate issue about that somewhere?)
Hopefully that helps someone?