,Just saw this thread. I'm a professional dog trainer w/30 yrs exp.
My advice is to correct any behavior that is remotely interested, let alone intent. The stare. The stiffening. Don't let it even get to the stiffening, nail the stare. E collar on at least a 5 out of 10. Dog isn't allowed to even look at Kitty, much less stare. Much less approach. I'd use a muzzle until you see some recognition of Kitty=NOMB (none of my bizniss). Play close attention and zap if you see anything longer or more than cat walks in room. Dog looks at cat. Dog should immediately look away, or if laying down with head up, should drop head to floor, WITH EARS AND BODY REMAINING SOFT. Not tensed. You can use a corrective sound - EH! or Cesar's SHHHT!
This isn't something to ef with, and a moment's inattention can have disastrous consequences. And there are dogs who will always be catkillers no matter what you do. But I've also known confirmed catkillers learn to exist indoors with cats, though not outdoors where the temptation is far greater.
You are looking to have your dog totally ignore Kitty. That's what you work toward.
I think the feral cat trapping idea is a very bad one, and I hope for both cat and dog's sake you don't do it. You have no idea if wild cat will indeed fight back, how your dog will handle it, and how he will remember the experience. Even if the cat opens a can of whupass on dear dog, will he learn to leave cats alone or will he accelerate into hating with a vengeance? Not to mention how very cruel that would be to a poor wild Kitty. Good Lord
Hope this helps. But if you still cant trust him after working at this, find him a dog-only home. Don't get your cats killed or maimed. I had a cat killed by neighborhood dogs a bit ago and it still haunts me.