Thanks for contacting me about this, and for your help in getting it working after our email exchange.
I’m responding here for anyone else who is looking for the same thing.
The automatic updates feature should be working since 2.46.2, and it looks like there’s no special integration required for ManageWP per their announcement below:
To enable MapPress automatic updates:
1. In the MapPress settings, mark the ‘automatic updates’ checkbox and save.
2. Enter your license key in the license field. Save the settings and make sure that license is valid (a green check mark next to the license key).
3. Automatic update should now work from the WordPress ‘plugins’ screen.