how do you score it.

It's the old 9-3 fielder's choice at 2nd base. No different (from a scoring perspective) than if he'd bounced to short and beat the relay at first.

Definitely a FC for batter in this situation because an out was recorded. Just an observation regarding the baserunning though, The runner on first probably could have been more aggressive since the first baseman ran to cover second base. Smart baserunners pick up on the fact that the first baseman decided the ball would probably drop and followed him at least 3/4 the way. In reality 3rd baseman or even pitcher should have moved to cover 2nd base so the first baseman could stay at his bag for a possible double play if the ball was caught. My thought was that the runner could have been more aggressive if first base had no fielder covering it for a double play on caught ball
Now it is entirely possible that the pitcher ran over to cover first base because the ball was hit that way and that freed the first baseman up to rotate over and cover 2nd.

Agree, except you have to be a little careful. I've seen a defensive team intentionally have the first baseman head towards second to coax the runner off first. They then back pick the runner with the catcher, who shows up out of nowhere at first base while everyone is looking at the ball in OF (including the first base coach).

Doesn't happen often and is usually done on a basehit, not the scenario described. In the case described, I agree that too frequently the runner is standing next to first base as the ball drops instead of being at least a third of the way to second.