Flight departs from US, but since EI are a Euro carrier, 261 should come into play.

Loads of revenue seats available on alternate routes, just no award seats. Sounds like they will be ok with opening up revenue seats for me.
However, no matter which alternative is choosen, there is no way to make it to my final destination within the EU261 time constraints, so I think I will be eligible for cash compensation.

Sounds like I would have to pursue EI for this, which is what I originally thought. Will likely have to fight, since they claim it is all BA's problem when I called them up earlier.