You should be able to use the REST API since menu items are just a custom post type in a particular taxonomy. There are easier ways IMO. You could create a custom page on the master site that only outputs the menu and nothing else. Fetch this page within an iframe on the slave sites. Or you could get the same data through other types of requests, for example by going through admin-ajax.php or admin-post.php. The returned content could be placed into the slave’s DOM through JS or jQuery instead of using an iframe.

There are pros and cons for each of these approaches. One of them probably makes more sense than the others. Even REST may make the most sense in some situations, though they are probably few. Personally, I would probably utilize iframe content if there were no strong reasons not to. YMMV