False statements are generally protected by the First Amendment.
If the video was an obvious gag or work of fiction, in which a reasonable person would understand that you were not truly endorsed, your false statement would almost certainly be protected by the First Amendment.
But many false statements are not protected, typically because of their negative effects on other parties. Classic examples include defamation, perjury, and fraud. A lie about an endorsement could easily fall on either side of that line.
If you are making the video in a context in which reasonable viewers would be led to believe that you truly were endorsed by that company, I could easily imagine a set of circumstances that would lead to liability:
- If you're releasing the video as part of a commercial endeavor, you're looking at possible claims for false advertising, unfair competition, and trademark violations (false designation, dilution, etc.).
- If your video is harmful to the company's reputation, you're facing a potential claim for product disparagement or trade libel. These are the corporate equivalents of the libel actions most people are more familiar with, and they generally require proving the same kinds of facts.
- A right of publicity/misappropriation of identity action is possible but very unlikely. Most states are very stingy about allowing businesses to bring these claims, so this one would probably be very jurisdiction-specific. Proving the case generally requires the company to show (1) that you made a commercial use of their identity, (2) that they did not authorize it, and (3) that the use caused economic harm.
Beyond those, there's a whole host of business torts that I could try to squeeze this into but that are probably less likely given the hypothetical you've presented.
I'd also note that your assumption about proving damages is not well-founded. Especially among the largest companies, businesses are very protective of their brands, and they invest absurd amounts of money to protect them. Pick any company with a high enough profile that someone would want to fake their endorsement, and I'd be willing to bet that their legal department could pretty quickly prepare a detailed estimate of the related damage to their brand. If a case went to trial, you'd normally expect to see an economist present an expert report calculating damages, which would begin (but not end) with any unjust enrichment you experienced because people thought you had the company's endorsement.