Dura ace pedals

I think it would be helpful if you told us how old the pedals were. I take it they are not brand new.

Also, how old is the bike, bottom bracket and chain set? Could be the bearing in the pedal. But this is not likely is the pedal is 'new'.

The only other suggestion I have, since you screwed the cleat, is check the bottom bracket.

There certainly is stuff to service on Dura Ace pedals, but it is rare to need to do it.

you would notice any play in the bearings by holding them and rocking, or attempting to rock them.

As said above, move the pedals to a bike that hasn't got these issues, to narrow down what is at fault. If the fault follows the pedals then they are the problem, if the fault goes away when they are mounted to a different bike, then something in the drivetrain is the problem.

Have you checked the chain-ring bolts are tight on the small ring?
Check the crank arms are tight, try rocking them side to side, and also look for play in the BB.

Have you tried other shoes on the "duff" pedals?

Guessing by "other bikes" you mean other pedals on other bikes. Or have you swapped the pedals over onto other bikes. In which case could it be the crank/BB or even frame that's the problem?