As far as I can tell, both LHCS and HST mark 3s are standard 415VAC 3 phase (which means you also have 3x 240VAC single phase where required), internal to the vehicles. The difference is that the LHCS have a motor-alternator set per coach to convert from traditional 750-1000VDC supply to standard 415VAC. Modern refurbs and retrofits can, of course, use a solid state static converter, thanks to several decades of advances in high power electronics, instead of a big spinning shaft with DC in one end and AC out the other. I wouldn't want to swear to LHCS mark 3s having only the motor-alternator hooked direct to the DC, but it certainly appears to be done like that to allow common/standard "off the shelf" industrial electrical gubbins.
HST trailers basically save the weight of the converter by running an AC bus down the train, and they obviously also have the necessary HST multi-working cables to link the power cars.
(The above is based on a lot of reading, including the specs publicly provided by some ROSCOs. Anyone with direct hands on, please feel free to correct me as needed.)
Going by the blog, that was a compromise made when rebuilding it. There's a blog article from the team which talks briefly about changing from 3 phase to DC for the auxiliary generator. I believe it was originally the same as the 43s for hotel power.
As for the question asked by the thread, sure, they should be able to be converted just like any other mark 3. In principle, it should be "simple and easy" (but that doesn't mean quick or cheap), unless they have an unusual peculiarity compared to normal mark 3s. Whip out the motor-alternator, extend the 415VAC wiring to come out both ends as a suitably heavy duty bus, then add the HST multi jumpers and bus. Oh yeah, and don't forget the overload protection devices sprinkled on as required.