How do I tone down onion sharpness already in the chicken salad?
As much vinegar as the salad will take might help a bit. Salt too, but your window there will be even narrower. Obviously, too much of either of those ingredients will ruin the salad.

Basically, I'm hoping for a bit of a 'pickling' effect. Onions become dramatically less sharp after even the briefest of pickling. So I'm hopeful that a reasonable amount of extra salt and vinegar plus time will help.

Of course, making a bigger salad by adding more of everything else is your safest option, but I'm sure you have already thought of that. Also, time alone will help some.