Look at crosmans co2 offerings, no cocking required. Like everyone else has said you need a rifle for this, shooting birds in the body will lead to slow excruciating deaths, it needs to be head shots, even an expert with a pistol is going to have trouble doing that
Do not sell this man any air guns no matter what he says
Obviously the laws in the USA are very different from our own, but I don't consider any pistol suitable for shooting pests apart from dispatching cage caught pests
I would advise you get some clarification on the details of the situation especially the shooting range & layout,
I'm guessing that his rifle is pretty high power hence it's hard to cock it, where a lower power rifle would require less cocking effort & might be an option for his wife,
another option is CO2 if it's legal over there, but whatever the choice it needs a shoulder stock so it can be aimed correctly.
might be better off with a small calibre shotgun, 9mm (garden gun) should do it, don`t knowabout the suppy issue in the states though?
If it shoots BB then it's not really up to the task, your most affordable options for the US wuld probably be a Daisy Powerline 880 or Crossman, they'll pack the power needed for a clean kill and can be found in places like Walmart.
I think Pyramid Air has a forum, that might be a better bet for getting the answer as it's US rather than UK as mentioned we can't shoot Starlings here and Grackles are not native.
You'll Shoot your eye out Kid
As well as the Crosman’s mentioned, look at the Snowpeak XS501 in .22. They have power adjusters when you remove the stock, (only 2 screws). They hold twice the CO2 that the Crosmans do. Look them up on the USA forums.
Why is everyone mentioning the USA?, the OP is in Moscow - pretty sure that's Russia.