Or go credit card touring, as I did a few summers back with a mate.
Day 1:
0800 start
London to Newhaven (follow the FNRttC route).
Newhaven > Dieppe (let the ferry do the work).
Night 1:
Any one of a hundred hotels in Dieppe harbour.
Day 2:
Exit Dieppe on the Avenue Verte, aiming towards Beauvais
Night 2:
Either Beauvais, or the first decent looking place after your legs get tired.
Day 3:
Beauvais (or thereabouts) to Paris. There are a few nice approaches. We went via Chantilly (which was beautiful, but cobbled and considerably off route).
Night 3:
A bar in Paris, with the bike stuffed into your hotel room. Join it after midnight.
Day 4:
Paris hotel to Eurostar, and back to London by early afternoon.
With two sharing, I think it all came in under £200 each, incl. food, drinks, ferries, trains, accommodation, etc.
Andy.