Hi, I was chatting with Dave R last evening and this problem cropped up. Both of us agreed it's a tricky one.
But. How about treating it as a round mortice. Drill out most of the waste with something crude and fast. 1 1/2" flat bit say, then clean up with sawtooth or forstner bit, that way you have much less waste to clear. Of course you loose the centre once you use the flatbit. So.. start off with the forstner to say 1/2", drill out the waste, then return to the forstner to clean up using the outside of the bit for reference. This should also take some pressure off the dilling machine.
xy, after chat with Dave R