Best way to perform application level high availability authentication

If you just provide the app and not the hosting environment I personally would not bother with such a case. If there is no room in the budget to make a simple database replication or even a backup for that matter I would guess there is no budget in the project for such a function request (basic login / tickets via SQLite). If there is room in the budget I would advice to spend it on a cheap VPS for MySQL replication or something. You could then even use the 2nd db as a fall-back connection so customers can access everything, not just the login/ticket part.

I would use the database error page to inform the customers that it is currently offline and that they should call a number or send an email or something. Be sure to mention that it is a hosting related error and that they should call there hosting provider to have it fixed ;-)

But if you are really worried about MySQL going down by itself why not migrate the whole project to SQLite? That would reduce that single point of failure (although technically you would create another one).