The MapPress settings only lists categories and tags, but when adding a new filter the ‘filter type’ dropdown should list all taxonomies.
There are a number of settings related to making a taxonomy public, not just the REST API. An easy way to test is by using the CPT UI plugin to add the taxonomy:
https://wordpress.org/plugins/custom-post-type-ui/
The default settings will make the taxonomy publicly available, and it should show in the MapPress settings.