If you can cook them before rigor mortis sets in you should be OK. Once rigor starts though you'd be best off waiting until that relaxes to cook them.
I've had rigor set in before I could get them out of the cone and to the scalder (under 5 minutes) and I've chickens still not be stiff an hour later.
If you want to cook a couple that day I'd hand them off to the chef the instant you're done gutting. And I'd pick a couple that went especially fast from cutting the neck to plucked and gutted.
Note: I say this as someone who's never tried to cook a chicken the same day. Usually chicken is not something I want to eat after a long day of processing them, so the day or two of rest in the fridge is just what I always do.
No matter how gently you kill a chicken, there's likely some adrenaline in the chicken and it naturally decomposes in 24 to 48 hours which is why they generally suggest that you let the meat "rest in the fridge" that long. Not doing so, may affect the taste for the worst, but if you're BBQ'ing, and adding a lot of spice as well as smoke, it may not be noticeable.