As someone that has used a gun to euthanize animals from the size of newborn puppies to horses, IMO ... euthanizing a small animal( less than 40lb. or so) with a gun is harder than a larger one.(from the standpoint of making it instantaneous)
By "tricky' I meant They can move or jerk around faster and more unpredictably, increasing the chance of ruining the perfect shot. It's harder to get the perfect angle while immobilizing the animal yourself, unless you're using a pistol (and having an assistant is a HUGE NO-NO for the inexperienced). Pellet rifle wouldn't creep me out because the consequences of the shot going into the wrong animal would be far less.
I have never shot a rabbit except hunting so I'm inexperienced in that, but brains work pretty much the same way in different animals.
I'm almost certain that bore contacted to the head and into the braincase with a .22 would do the trick to instantly kill, likewise with a pellet rifle (except to the forebrain perhaps). Rabbit brains aren"t very big, after all, and bore contact would maximize overpressure inside the skull. Most probably, the overpressure without bore contact would be more than enough, so bore contact is in the "overkill" category. Destruction of the hind brain is ideal since it shuts off signals to the body causing instantaneous flaccid parylazation as well as stopping the heart and breathing. The overpressure provides a shock to the rest of the brain to take care of the "consciousness" aspect. However, I've never done this with a rabbit, opting instead for other methods.
If I had to kill a bunch of captive rabbits, I'd opt for the club, since in theory, it provides both partial internal decapitation and a physical shock to the brain. I also have never seen a kill board much less seen one used and that's the only reason I wouldn't use it. Since your dad's aim with a club may not be good, or if he's not comfortable doing it that's out. Salvo's broomstick method sounds really good, but with the kill board, the weight of the rabbit is working for you.
Your dad's comfort/confidence is what's important in choosing a method.
for those disturbed by mentioning puppies ... mother dog died after birthing them because of heartworms and it was long before there was a cure for them. I used a .22 rifle bore contacted to the head on them, but new puppies don't move around much in a shoebox. The dogs were owned by an idiot down the road. I got the job he supposedly "couldn't do" after he did nothing for her. Same with one of the horses, but a different reason and a different idiot. (I've gotten stuck with sucky jobs for no pay too often.)