So, you just barely got the tap started before it popped???
Anyway, if it's a regular carbon steel tap, just pound it out. Take a punch sized so it can fit in the "through hole" but smaller than the threads and break that sucker out. It's scary, I know, but you'll be surprised how easy it is to simply drive it out, snapping the threads right off the tap. AND, the threads in the lockplate may even be just fine.
Through holes are easy. If it was a blind hole.....
With those I have had to just heat and anneal it and drill it out. Not fun, but still doable.
Oh, and what drill bit did you drill your hole with? If it is even one size too small, you're almost guaranteed a broken tap.