In my opinion I would leave it as it is, it will be fine in a month - already cordoned off so the drunks don't try to bathe in it). But if one felt the need to give someone a job and spend a bit of money, then, yes you could.
Leaving the lawn to dry out a bit (a lot) and then levelling the furrows, maybe a bit of re-seeding, if required. Maybe prudent to ask a local councillor about it and then let us know the news, but to be honest there seems little point touching it when the ground is waterlogged. Best left a few weeks
Wet weather + big lorries on grass = mud. It is what it is. If the weather was dry it would be fine but you cannot stop it raining. The grass will grow back, no harm done.
"There is one good point - The way out to Teignmouth is clear, where there is plenty of free parking, some remaining shops, a good playpark and seafront walk that isent crumbling away and a coach park close to the action." Agreed the difference between the two is tremendous, with a little work Dawlish sea front could also be as clean and welcoming. The only similarities are the dog poo! ...
I personally feel it would have benefited Dawlish if that dirty looking excuse for a snack bar next to the beach was demolished and rebuilt. It looks awful, stinks of dog pi$$ and I pity anyone that eats from it. I imagine it's privately owned and in which case the council should step in and force whoevers hand it is to bring it up to scratch. All that cracked and half repaired concrete does ...