I think if they had to argue it in court, they will present the cache as an irrelevant technical detail, just like how fleeting copies of frames from a DVD are

I think if they had to argue it in court, they will present the cache as an irrelevant technical detail, just like how fleeting copies of frames from a DVD are stored in buffers and decoded when playing the DVD on a traditional player. The Doctrine of First Sale allows people to rent out DVDs, and people in lawful possession of a physical object like a DVD to watch them (but not make a copy); if the transient data stored in DVD players for the purposes of decoding and playing a DVD don't count as copies. US law explicitly gives the copyright owner exclusive rights to authorise who can "display the copyrighted work publicly" - but if it is only going out to one person, I suspect that they might not be able to prevent it.