First, you don't need to pre-cook the chicken, you can raw pack it and process it, and the process time in the canner is the same, so the overall process time is greatly reduced.

Assuming half a chicken per quart (unboned), or 1/4 chicken per pint (unboned), 3 chickens would do a short load in a canner. My canners will each do 6 quarts at a time, roughly two hours in the canner (15 minutes to get it up to pressure, 90 minutes at pressure, 15 minutes to let it come back down to ambient pressure). Figure half an hour prep (cutting up the chickens, stuffing the jars, putting on the lids, etc). So you could do 6 quarts in an evening easily.

But it's really a matter of priorities, and it sounds like your schedule is too full to mess around with it, so probably better for you to just buy what you need at the store and forego the trouble. If all you have is an hour or two a month, that's not going to work. It won't be as healthy for you or your family, but that isn't always the most important thing.