Good luck with the trap.
I wouldn't expect it to work on coyotes. Maybe a coon or possum.
This years pups should now be out and about, hard and heavy. One of them may venture in to the trap after a couple of days when your scent as dissipated. But the bait will likely be all gone by then. If you can use a liquid bait/attractant and wear rubber boots and gloves, you may be able to fool one.
As for baiting, it will feed them. You just won't know exactly when!
Coyotes, especially during the summer heat, are much more active at night. They would likely hit the bait pile around 2 or 3 am.
If you do anything to their den, even get close to look at it, them mom will relocate them immediately. They will usually have several den sites in the area just in case.
Coyotes may be the most adaptable and most intelligent predator that our Father created. They will survive!
That being said, I would try a few different ways to get rid of them.
1) (My Favorite) Do a bit of observing of the coyotes. If they are denning there by the water then they are likely traveling a short distance to a hunting ground or for territorial issues. For what ever reason, there will be trails that they will use to get to and from.
Get a predator call and set up on on of their path. Find a spot where you can see but can't easily be seen. Preferably with the sun at your back and a cross wind or the wind in your face is a must.
Sneak in to place before daylight or about an hour and a half before sunset. I would sit perfectly still and quiet for a while hoping for an ambush. After about a half hour I would lightly blow a distress call hoping to bring them in.
Blow if for a little over a minute and then watch in quiet for about 7 to 10 minutes. A feather tied to a limb with fishing line about thirty yards in front of you will work wonders here.
2) You know where they live! Set up like a deer hunt either at dusk or dawn and wait for them to come or go. They should exit the safety of cover and pause to survey the area and present a shot.
3) (Dangerous in dry conditions) Put on some old clothes and carry a small shovel, some lighter fluid and the .22 and go into the briar patch to find the den. You want to be as quiet as possible or you will flush them out. If you can bring a handful of soaking wet green grass that would make things a bit safer and effective. You don't want a large fire. Just something that will produce a bunch of smoke at the entrance of the den. If you can get the smoke and fire INTO the den the coyotes will bail eventually.
now getting a good shot with the .22 will prove to be difficult. A shotgun like big al hunter suggested in each case would be best.
There have been MANY coyotes killed with a .22LR. I would suggest getting some CCI Velocitors or Stingers. Something with a bit more ump! Make sure to site in the rifle with any new ammo. To get a recoverable kill so you can take pictures to show us
, I would still try for a head shot. If that isn't possible go for the heart. Now keep in mind that the smaller coyote has a smaller heart and it is a bit forward then most game animals. They also don't have much of a shoulder blade. Thus the ideal shot would be at the top of the front leg just below center of mass. A shot NEAR the vitals will most likely kill it, you just may not recover it.
In the event you are successful, we want to see pictures!!
Good luck brother!