Since the 1960s we have messed around with modifications to aircraft to hook things. I will admit, they were mostly at low speed and low altitude, but I should think shooting it down would be easier. If the technology is solid state, data could survive a hard landing. I suppose they would want to do it in a remote area but closer to a suitable base to retrieve it from. Unless, of course, we are now a wholey owned subsidiary of China, and don’t want to actually retrieve it.
Was also demonstrated in a James Bond film: 
One would think that the USAF or CIA brass (at least the ones not on the ChiCom payroll), could resurrect that system, catch the balloon and drag it out to the ocean, release it and shoot in down. Of course, if it has a nuke to cause an EMP, all bets are off.