They can't (because LCoE is a per-project or per-technology measure, not a grid-wide measure, for a start).

UK energy is expensive because we have gas-linked wholesale pricing. That's nothing to do with the true cost of renewables. I'm going to go out on a limb and say they're being disingenuous.

(Gas-linked pricing was implemented for sensible reasons, but I don't see how it continues to be tenable today).

Yes, that's what I thought, but wanted to give them the benefit of the doubt.