Is the developer asking for a WordPress admin account, or some other server admin account?
Depending on where the plugin is hosted (such as the WordPress plugin repository), you might be able to find an older version; there are numerous pages that go through the various ways you can do this.
An opinion: as a developer, if a version of a plugin stops working and it appears like it might be a bug, I don’t treat it as a support case but as a QA case. This comes with a different set of responsibilities for both parties: for the issue reporter, the primary responsibility is to provide enough information to reproduce the issue (but to weed out irrelevant information). If you can do this, the developer should then be able to implement a fix in the next version. In these cases, the developer never needs to touch anything on your server, which is what makes a case more of a support issue. Also, debugging generally requires having more control over a server than you’d get with just a WP admin account (though server access can be helpful in figuring out how to recreate the issue). As for how this applies to you, you can see if the developer is open to this route.
Another option (if the developer is only requesting a fee for screen sharing and not to work on your server) is to set up a new site so that you can safely give them access. In particular, set up just enough to recreate the issue (this is also a way to fulfill an issue reporter’s primary responsibility) and don’t put anything on the site that could be used to compromise another site (e.g. don’t reuse passwords from other sites).