I laid a lawn a few years ago, I never went to the extremes of the chap you had. When we moved into the house, the back lawn was awful (in fact it wasn't a lawn at all, just what looked like a builder's yard). Anyway, I dug it over by hand, roughly, then hired a chap to rotivate it. I then raked it, and then did a choo-choo motion over it. It took a couple of hours, but I got it flattened. I then ordered the turf, and laid it brick-like fashion, and I had a perfectly laid lawn. In your case, especially whilst its damp, try obtaining/borrowing a heavy roller and get the kids to use it by running it up and down the lawn over a period of a few days or more. This should do it. Me and my brothers always did this for my grand-dad (who was a gardener). The bigger the garden the better.
... and they tell you to keep off a newly laid lawn ...! I hired a roller and that was just about hopeless. I filled it with water - though I suspect it should have been sand. My only fear is that the lows in the lawn, being softer than the highs, are likely to get deeper with the pitter-patter of tiny feet zooming across it.
I think I've decided to wait a couple of months - say until September - and then slowly top-dress while there is some growing space between then and November. Does anyone know of a supplier of good topsoil ... or give me the ideal formula for this type of dressing so that I can go to a local supplier and ensure that I get the best stuff -- rather than graded builder's refuse??