I know that grounding into a double play with nobody out, with a run scoring from third, does not reward the batter with an RBI.
What if, however, the bases are loaded, nobody out, and you hit a fly ball. Runner on third tags and scores. Upon the throw coming in from the outfield, the cut-off man intercepts and throws to third, cutting down the guy tagging from 2nd to 3rd (hence, a “double play”).
Does the batter still get the RBI in this scenario?
Yes, the “no RBI for a double play” rule only applies when you ground into a double play.
More precisely it only applies when you ground into a force double play or a reverse force double play. A force double paly is the standard one with a runner forced out at third or second and the batter doubled up at first (I assume you could also force out runners at second and third) A reverse force double play usually occurs when the batter gounds to the first baseman who steps on first then throws to the shortstop who now must tag the runner coming in from first. You do not get an RBI in this case either.
If with runners on 1st and third you grounded into a double play by having the first baseman force you out then then trapping the runner from first between second and third after he’d rounded second, you would get an RBI even though it’s a ground double play. (BTW I’m pretty sure you are not charged with a GIDP in this case as well).
And in the previous answer with the outfield fly double play. You get credited with a sac fly as well.