If it works locally but not on a remote host then first thing to check is whether the file is being correctly located, is it? Also drop back to default BP theme does Ajax now work on the remote host?
tbh it took me seconds to see that nearly all WP library scripts were 404’s you need to establish what’s going on with your script includes, it’s not an Ajax issue it’s more fundamental than that, essentially JS isn’t running on your site.
JS isn’t a server side process it’s client side, so not a question of whether it’s working, all the server does is send the js file on request the browser parses the instructions within the file. You want to be checking your paths carefully and general install.