10/4*: Put my opponent on the bar, leave 15 shots.

10/8 7/3: Safety 1 checker, also leaves 15 shots, doesn't put my opponent on the bar.

The wrong play costs 5 cents, so it's a fairly significant error. I can't figure out why the right play is right.

I opted not to hit. I figured they might roll a large number and get out of my hair, and I'm leading significantly in the race. Turns out the right play is to hit for some reason. I don't understand why, it's not like I have gammons to fight for!

I calculated the same as you, 15 shots for each. I also figured it was best to safety the checker since the shots are the same. I can think of a plausible explanation for the hitting move:

After your move you'll be 36 pips up in the race. If you don't hit and the opponent rolls 55, or 66, they'll have a 25% or 30% chance of winning, respectively. Notice they don't have to roll it right away; they can get past your checkers, roll those doubles later, and still have that chance. So you're giving the opponent >1/18 chance to have an unwarranted considerable chance to win, one that you don't give with the hit. Also, 65 and 33, three numbers, give them a 10% chance of victory.

If you hit and your opponent dances, your win becomes even more assured, given that with so many spares on the 3-point, you probably won't be forced to leave a shot before the opponent re-enters from the bar, and even high doubles from the bar straight away after they dance will only give them a ~10% chance of victory.

I first crosschecked the evaluation (you never know when you stumble over a bot glitch) and both bots agreed on the amount of error. I thought ah the gammons... but it's a 1-pointer....... but when the bots agree there must be some reasonand I guess I found it.
If you play 10/8 7/3 what happens if you roll 1-2,1-3 or 1-1? You give him another shot. If you hit and are not ht back you can cover.
Very nice position!!!

I figure that since you have a lot of wastage, this race still isn't a lock. So if it's the same number of shots, might as well try to improve the race.

I thought so too initially, isn't it better for us to have the checkers further ahead? It might seem so, but see it like this: in a no-contact position, if you were 30 pips up and your home board looked exactly like White's, you'd be 30 pips up (only the dice will determine the outcome). BUT you're 30 pips up with a home board that has more wastage than the opponent's, so by logic, you cannot be in just as good a situation as the first case. In practice, it's the equivalent of having a board exactly like White's, but being fewer than 30 pips up. So the wastage you have diminishes your real pip advantage.

Compare, for the same 10-pip difference:

If black didn't have to adjust his and white's outfield checkers to equalize the pips, then of course he'd prefer the second option. But alas, we're comparing equal differences of pips, so that cannot happen.