I'm on minocycline as well. I wouldn't be too afraid to give it a whirl. It's helped me. If you're worried about side affects, popping a pro-biotic can help, especially with digestion issues. It's not a long term fix usually... but hey, if we can get away with it and it helps, why not? Gives you a breather for topicals and lifestyle changes to kick in. Or even better, just for one's hormones to settle down on their own if that's the root cause for you.
Aw. That's the worst, Tyrean. I'm just getting over such a cyst myself. The inevitability of the forming stage is the worst!
On the topic of BP, yes, a lot of people use it topically. I just started a few weeks ago to use a wash (was using some heavier stuff before that--tazorac), but applying a standard BP 2.5% cream like you mentioned should be fine, like Dan recommends after cleansing. Will it help a cyst like in this scenario? I'm not sure. I used a washed every night on this guy and I'm quite tempted to say it helped, but obviously not as drastically as to be readily apparent.
A product I found helpful for treating cysts is the topical anti-infection cream Aczone. Unfortunately it is prescription only, I believe. But it's worth a google search. It seems to take longer for the cyst to break and clear, but it is much less angry and after even as short as a day of topical Aczone becomes hardly painful at all.
A note on the tea tree oil--beware, it can definitely hurt. NEVER use tea tree oil at 100% strength, always dilute. It will burn you something fierce! Believe me I know from personal experience. Here's a link I found real quick on diluting. Hope it helps!
I know it seems like that sucker will last forever, but it will pass, and things will get better. Keep your chin up. We're rooting for you!