Since I agree that the market might be too thin for nonstop Basel - London, what could be done is your proposal in Reverse: Two sets to Lile Europe, with the Front set going to London (Lille could be a technical stop only) and the rear one to Brussels and maybe Amsterdam. SNCB and NS International might even be open to a Cooperation.
That certainly would be a sensible option, as long as the Home Office can be satisfied that passengers in the two sets can be kept separate and that the London set can be kept secure if the train stops to pick up set down Schengen passengers at say Lyon or CDG.
Another option could be to have two trains with two units. One train from Basel and one from Geneva, each with a London half and a Brussels/Amsterdam half. Assuming you could find a suitable location you could do a shuffle of halves and send the two London units off together and the two Brussels/Amsterdam units a few minutes later.
From the Home Office's POV I suspect the risk of incorrect door release would be too high - it would be 'incorrect' not for safety reasons but purely to satisfy a border arrangement so it would be a new type of railway procedure for which there likely won't be strong mitigations. And I doubt the Basel - London market is thick enough even for a 200m unit.