Batter-Runner Interference after play at Home

[NYBAREF]

Here is the play.

Runner on 1st and 3rd, 1 out.

Batter hits ball to Pitcher, R3 breaks for home. Close play, but the Cather misses the tag, and R3 touches Home Plate. Cather steps inside and fires to first to try and retire the BR. In doing so the throw hits the BR in the helmet while the BR is entirely the the left of the running lane.

Questions: Should interference be called on BR?

If so, shouldn't all runners be sent back to original bases, as not all runners advanced at least one base prior to the interference?

My thinking is that the run should count because the defense had an opportunity to retire R3 PRIOR to the interference.

Would the ruling be different for OBR, NCAA or FED?