Exactly what kind of tub stopper do you have? Could you post or find a pic of yours? If we can get you access to your drain directly then maybe you could use something like this to clean out the drain:
Otherwise you'd have to go through the overflow and use one of these, which all plumbers should have. Here's what the set-up would look like:
Please do me a favour though. If you do end up calling in a pro please tell them about your use of chemicals in the drain. That way they can minimize the potential splash back and avoid injury.