specify the location of our externsions somewhere other than the Application Support folder
I think you can do that already by replacing the PopClip folder in Application Support with a symlink into a folder in your Dropbox folder. As you say, it would need to you restart PopClip on the other Mac after making a change on the first. Not officially supported but I think a couple of users already do that, might be worth a try.
That said, for a proper syncing solution I would want it to be more robust than that and also sync the settings and the order of extensions.
Welcome to the forum, @rkaplan 