Aimee, your dog does stop/sit if the rabbit does not run but does not stop/sit if the rabbit does run ….is that about correct ?
If it is correct then it isn't getting hold of the rabbit that is important to him , it is the chase.
If you can try getting a large rabbit skin and sew it or glue it around a fairly large rubber ball or maybe a cricket ball. Sew or fasten two rabbit tails to it at opposite ends of the ball and DO NOT EVER LET YOUR DOG SEE IT OTHER THAN WHEN YOU ARE TRAINING HIM.
Carry the ball with you in a small gamebag , put the dog on a checkcord with a toggle on the end you will hold , then be ready to throw the ball, or have someone - - - preferably male --- throw it for you while the dog is hunting. 9 out of 10 males throw far better than almost all women ! The throw should be made as a bowling motion to send the ball over the ground, not through the air.
This needs planning beforehand, the ground must be flat enough to allow a ball to roll and it should be on a downhill slope. It can be very difficult to time the throw correctly for the dog must NOT see the throwing motion. Perhaps the helper could hide behind a bush or a tree ?
The ball will roll and tumble over the ground and if the dog is fooled by it (most are fooled if the throw is done at the right time and is not seen by the dog) it will chase the ball. That is your chance, take the dogs head off ! Give it a damned good yank on the cord ! Don't mess about or the lesson will be diminished. Then, still holding the cord, walk along it to your dog and make him sit where he should have sat when he first saw the "rabbit" run.
Repeat the lesson after a few moments in a different place then stop at that for the day. Too many repetitions of this, done too close together would possibly greatly diminish the dogs willingness to hunt and would increase the chances of the dog seeing the throw . Do not let the dog see the throw ! Arrange a signal with the helper in advance of these hunts for you must decide when the ball is to be thrown. Don't throw the ball really close to the dog or the dog will catch it before you can pull it back .
Never allow the dog to closely examine the ball and the dog may continue to believe it saw a real rabbit for quite some time ....long enough to train a stop to whistle and stop to the sight of a "rabbit" running. Leave the dog at the sit after each lesson then walk out, pick up the ball yourself and put it back into the gamebag.
I have used this method on fairly steep hills with a rabbit skin and tails tied with string around a turnip ! I also used it on steep hills with a checkcord on the dog ..... and a checkcord leading to the turnip ! The whole hunt was arranged in advance unseen by the dog . I tied one end of the turnips cord to a small stick that I pushed a short way into the earth --- just deep enough to hold back the weight of the turnip I put in place directly behind it. The other end of the turnip sticks cord was left lying on the ground in the direction I'd be hunting the dog down.
The dog was cast off to hunt, I'd pick up the cord and pull the little stick away , freeing the turnip to roll off down the hill. I'd then yank the dog to a stop if it began to chase .

The things I've had to do just to train dogs !
Getting the timing of the throw right is very important if you try this Aimee. Do a little practice session without the dog being present first if you decide to try this Aimee and don't mess about trying to be nice to the dog , take the chance when it presents itself and give the bugger a "good 'un !"
There endeth the sermon for today on how to do it old fashioned style !
I think you are very fortunate to live in an area with rabbits on the ground , it means you get far more chances than most of us do to train steadiness to fur.
Wullie