I don't doubt that's exactly why they're so furious that a second round of consultation might be on the cards jhuizinga. They know it's going to play out in public rather than in private and they know that even people who support them but didn't really engage fully will now be watching a much wider, more detailed and two sided debate.
They won't get away with passing everything off as "transphobic" for much longer, I think even they realise it's starting to sound ludicrous. They also won't get away with their lies about "GRA not having any impact on EqA" because the conversation about overlaps and conflicts is becoming much more mainstream. Nobody can answer simple questions about how any of this can be applied in practice. They're also trying hard to push the "what is trans anyway?" question back in its box.
People are paying way more attention, detailed scrutiny is afoot and they don't like it one little bit.