Your suspicions are completely correct. Worse, the indemnification of others is only good as their assets and their willingness to provide a defense. Since the copyright to the other videos was established at their creation, whether or not the authors filed for copyright registration, you would need a consent or a license to use them under the circumstances you describe above. A lawyer for the business would help all concerned.
The above is general legal and business analysis. It is not "legal advise" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
YOUR THOUGHTS ARE ON THE RIGHT TRACK. CONGRATULATIONS.
There is only one exception and that involves getting permission from every one of the brands you intend to use. Highly unlikely, So, don't do it
USPTO Registered Patent Attorney, Master of Intellectual Property law, MBA I am neither your attorney, nor my answers or comments in AVVO.com create an attorney-client relationship with you. You may accept or disregard my free advice in AVVO.com at your own risk. I am a Patent Attorney, admitted to the USPTO and to the Florida Bar.
Yes, that would violate the copyrights in the videos they want you to composite. You could (and should always) have them agree to indemnify you, but that is not sufficient to protect you. It will not stop you from getting sued, it would just allow you to sue your client to get the money to pay the judgment and/or defend yourself. Obviously that can go wrong in all sorts of ways, including your client going bankrupt or disappearing.
You should have a standard contract with your clients that addresses indemnification and other copyright issues.
As another attorney has said, you are correct to be concerned. You would be the one to stand in front, so to speak, because you would be the one actually creating this composition. One could argue that each of your uses of other peoples' work was a 'fair use". Still, that and about four bucks will get you a coffee at Starbucks[tm].
The better tack would be to create a parody of each such video, although that is still not free of danger.
While I normally recommend seeking a license from the copr. owners, they may not wish to license being satirized, but you can try. Maybe half will go along with the gag, especially with credits at the end, and that may be enough to make your point, or your client's point.
Information here is general, does not create a lawyer-client relationship, nor a substitute for consulting with an experienced attorney on the specifics of your situation.