Another comment [0] pointed out that a version of this was already tried, and it was found to still be a public performance. Not sure how much the details of the technologies used would affect the ruling, but probably not enough.
> Not sure how much the details of the technologies used would affect the ruling, but probably not enough
Right. Aereo notwithstanding, one way around this might be to set it up like MP3Tunes[1] where you're a specialized digital locker service. The "fixed" "tangible medium" should originate with the customer, and a transfer from the customer-controlled copy to the business should be involved (rather than the other way around). With a large enough physical presence, you could get this down to pizza delivery time frames and/or Redbox levels of friction.
[0] https://news.ycombinator.com/item?id=32645286