I'm not familiar with U.S. laws/government, but I would imagine a law would have to be passed to allow this, before the armed forces can even start considering this and regulate/specify anything, because I would imagine that the rules on corporate sponsorship are pretty tight already and not include exceptions under which sponsoring the military could fall, for obvious reasons. Or am I wrong?
Also, I'm thinking in such a setup, awarding a sponsorship contract to the corporate sponsor would not be as easy as in the corporate world, and be more like government procurement. So even if you could set the rules in such a way to favor say Boeing or LM as sponsor... if they are outbid by a US subsidiary from whatever undesirable foreign company... by law they'd have to accept or get sued or fined... etc, etc. So I don't think they can regulate it easily as you say... they may still end up with something that's objectionable in every possible way.
We're only hearing about this sponsoring to deal with the current situation... but I don't think they're capable/allowed of accepting such a solution short term. So then it doesn't help in the short term, and there's no need for it long term right, because:
The current situation is not normal, the cuts are made everywhere except the real costly things because they are short term, we are not talking about structural spending cuts here, right? The US Thunderbirds annual budget is reported as $9.75 Million (mind you, taken from a news article on the cancelled season, may be lower than actual savings made by cutting the TB altogether, I don't know). The current USAF budget is normally about 140 Billion! Relative to the budget and all other spending plus potential areas of cutbacks (*cough* procurement, overseas bases, combat force), that $10m is a small price to pay for public relations and recruitment benefits it provides, so it can be (and I think, will be) easily justified. Because even if structural cuts are made on the PR and recruitment budget, it would not drop to such a low level where that 10m cannot be afforded... they've already cut the single-ship demos.
My point is: it's structurally not necessary to take a tricky sponsorship route, and airshows are still a very effective PR/recruitment tool. When automatic cuts don't apply, and recruitment needs go up again, and it will sooner than later, then there's no problem with funding the teams. It's only affecting the teams now because of the current forced cuts, that doesn't mean they are threatened in case of structural cuts. Unless they'll be really drastic. But even then... the DOD's budget has been 2x that of FY01, that's not just inflation in personnel and procurement costs, those have been and will be continue to be met by base closures and force reductions, right? It's been due to the wars, and all the "overhead" that came with it. They had the teams and singleship demos flying in FY01... no corporate sponsorship required.