No, HC-06 is a 3.3v device. The breakout board is made for 3.6 > 6v, and uses on-board regulators, hence the 3.6v. You can get an HC-0x on a 3.3v board, and one that is switchable 3.3v or 5v. Another and, in your case perhaps better, option is to simply get a bare HC-05, no breakout board, and share the supply to same with that for the Pro Mini i.e. a properly regulated 3.3v.
Probably not such a good idea, and I note there is no comment on monitoring the mA, which is probably more to the point. Check your data sheets. If you wire up the supply to Bluetooth in the proper manner, you probably won't need to check anything. Needless to say, "proper manner" also means adequate supply. 850mAh sounds a bit mean, but maybe you are already abreast of that, and this is just an exercise in short-term portability.