I am wondering if anybody knows it is possible to enable remote access to mysql on sharing host.
This should be possible via your hosting control panel (some hosts will not offer it though). Other than that, your best bet is to contact your host and ask for assistance.. or plea for a bit of understanding.
MySQL over the internet isn't the best for performance
Quite true. Given that Google now takes website load speed into account, it may be yet another reason to consider getting proper hosting for the site, if getting traffic to it is important.