I think I finally figured out what happened here. I've been using Godot v3.4, but when I tried the Mac export I downloaded the latest version (v3.5, which was released a couple weeks ago). There was a slight change to how it handles inputs that caused a problem with the UI buttons. One extra line of code, and it's fixed.
As it turns out, v3.5 also fixed the issue that was preventing me from code-signing, so I should be able to push updates and troubleshoot issues for the Mac version from my own device now.