No, it won't fry, but it won't power them either. The Core battery box is what the TA4F plugs into. The mics come out of the TA4F and also the 1/8" plug that goes to the recorder. If you don't have the TA4F plugged into the Core BB, there is no connection between the mics and the 1/8" plug.
In
this thread acidjack mentioned a "cap" to go on the TA4F (mini XLR) to allow the use of other battery boxes. I PM'd him about it but haven't heard back. Chris Church also said somewhere don't do it because the polarity is reversed. But as long you figure out what each connection in the TA4F/Core BB/Church BB do, it should work fine, right?
Core HEB: