The regular sleeper cars on the Caledonian sleeper have 10 rooms containing 20 berths, the accessible 6 rooms containing 11 berths. St Pancras platforms are apparently 455m (I'm going to assume major European terminals can cope with this, for non London UK stations only Edinburgh Waverley probably can) 22.2m vehicle length allows 20 vehicles, one will be a locomotive so 19 vehicles, for practicality lets say it's 18 cars arranged as 3 groups of 6, so Club, Accessible and 4 standard. 273 if all rooms are used at double capacity. 6 sleepers from London to various Euro destinations is 1638 passengers. Easyjet A321s carry 235 passengers, which is 7 flights for the same number of passengers.
I want to say this can work, but it really, really doesn't unless we build a huge amount of infrastructure or tax flying out of all recognition, basically 2 planes, for the length of flights in question can move the same number of people. Another plane can do a couple of trips Edinburgh/Manchester to Paris/Brussels and match capacity there.