Nice idea, OP, simply not possible.
Firstly any rolling stock which uses the Channel Tunnel, has to meet incredibly strict fire safety regs, which is one reason LPG cars are not allowed. A Eurostar train is actually two halves of a train which, in the event of the failure of one half, will permit the other half to rescue it out of the tunnel. Passenger carriages have special fireproofing too.
The freight and car shuttles have a 7500hp loco at each end (9400hp for freight shuttles) and the loco has to be capable of hauling the whole train, should a loco fail.
So to have a car from Manchester moved by rail, through the tunnel and be transported to Brussels, Berlin, or wherever, it would need the same kind of wagons used already, plus passenger accommodation, as it would be unrealistic to sit in the car for 5-6 hours.
Problem is, such suitable rail vehicles would be too large for the UK loading gauge and need expensive infrastructure improvements. Best we can manage is 9ft6in containers on wagons, and a max width of 8ft6in.
There have been newspaper reports of sleeper trains from Edinburgh, Manchester and London to European Capitals, but a lot of this is pure fantasy as the same fire regs apply, and there are no such vehicles in existence. To design and build them would be very expensive and have a questionable benefit, when it would be easier to catch an existing Eurostar from St Pancras at around 1900, have dinner on board then switch to a European sleeper train (whose network is expanding) in Paris or Brussels, for the rest of the journey to wherever.
Back in November 2010, Deutsche Bahn (German State Railways) brought an ICE train to London, as a taster for a passenger service it was proposing to Frankfurt, and the places. Has never happened, too many issues, and unlike to.
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