Here's my take.
Lucy is an incredibly GOOD DOG. Yes, she growled, snapped, lunged at the cats. But she could have bitten them. Lucy is warning. In my books this is not aggression. Aggression is a bite which to my mind Lucy is clearing avoiding. If the cats don't get the message though, Lucy might have to escalate to a bite in order to get her message across. For me it would be the escalation that needs to be prevented.
How long have you had Lucy? A dog introduced to a new household needs time to settle in. Existing pets also need time to realize what may have been acceptable with a previous dog might not be with a new dog. Personally I would not expect this period to be less than several months.
What I would (did and do) do. The cats were there first so they get fed first. Lucy has to wait, away from the cats, till they finish before she gets fed. The cats have their own feeding place and if they leave their dishes with bits of food left in, as cats often like to do, then the dishes are put up out of Lucy's reach.
Then Lucy gets fed, in a separate space away from the cats. I'd put her food down, watch from a short distance away while she eats in peace (if she's part Lab this will take, oh, about 2 seconds, LOL) and then remove her dish too. I would use NILIF with Lucy. Always. Short easy cammonds to start with, working up to sitting on her mat somewhere for as much as minute. Nosy cats would be shushed away by me. Both cats and dog deserve to eat their meal in peace.
Till Lucy has been with you for much, much longer I would not even try treating them while all together. I'm sorry but I definitely would NOT do what you did last night. To me it is unfair to feed the newcomer first, as I said above. It is unfair to taunt the dog by having her near the cat's food and unfair to the cats to have them threatened by the dog. It is unfair to correct the dog because at this point I don't think you can be sure the dog knows the order of things and this is a harsh way to treat her. I'm not against a well place positive correction but in this case what I'd use is a negative punishment of simply being withheld from the room while the cats eat. It's the same difference to you time wise.
Also consider this. Feeding cats first, Lucy removed, then feeding Lucy is a clear indication and positive reward to Lucy for allowing the cats to eat first. CAts finished, YAY, now Lucy gets her reward, her supper. Low key though the YAY is only in your mind, no excitable exclamations.
I've rambled a bit. If my ideas interest you and are not clear please feel free to ask. I have to admit, none of this worked with one new dog. It worked with all the others but one rescue we had to always keep separated at feeding time. You never know what's been in a recue's past. It's my belief that some things just aren't worth the effort and for us, with Whisper, separation was the easiest on all of us. Now, with my current dog well established, I can feed and treat all at the same time.