I believe that some do. And some work the Sleeper. There might even be some overlap between the two sets, but it would be by chance rather than by design.
Most of the 92s (including a good number of the "Sleeper" locos) have TVM - and I don't think it's a huge job (for 92s) to get it sorted/certified
if they wanted to use it. However, I can't see Serco wanting to start their premium London<->Scotland service with "guests" having to make their way out to Stratford, though. Doubt it would be a realistic option.
The Serco/CS Chairman actually spoke about the alternative options in an interview when they won the contract - his favourite was St P - I think more based on the platform length/grandeur of the building as a starting point for the service. As for the various reasons noted, though, it may not be all the feasible.
As
@captainbigun says, it's most likely Euston until they cannot use it any more, then Kings X - probably with split trains as happened last August BH.