Sqid, do you have the freezer space to freeze the flour? If you freeze it long enough, that will kill the flour mites.
I don't think shaking up the sealed container will make any difference.
In my experience, sealed containers make very little difference for flour mites. The eggs are almost always in the flour when you get it and, given enough time, sealed container or not, they will hatch.
If a sealed container does, indeed, buy you 4 weeks and you can store it that long, you're in luck. If, on the other hand, you're stuck with 1 or 2 weeks of storage, that's still, imo, going to be a problem.
Imo, if there was ever a get out of jail free card for using bromate, it would be living in a country where flour can't be naturally aged properly. I'm just saying.