Inner Circle
I think if the bait was good enough, they would. If they thought the Inner Circle or even the Dark Lord himself was going to be there, unprotected and unaware, they may well believe it worth taking the risk. Though I will admit that there may be better suggestions for bait.
Podmore must be more clever than he appears. While I do not have the highest opinion of the DMLE but he could not have lasted as long as he has without possessing some skills.
I agree that reducing their effectiveness would be useful. There are ways it could be done though it would likely need to be handled delicately though.