What does the Mad Hatter say to Bayard?

It's like you assumed: he says "Downal wyth Bluddy Behg Hid!".

It means "Down with the bloody big head" or as the source below quotes it: "Down with the Red Queen, the slogan of the Resistance." (source)

It obviously means that the hatter is hinting Bayard that they are on the same side.

The only thing that might confuse people here is when people assume that Bayard believes in red queen's promises that his family will be released. But I don't think people would still continue to assume the same way after that scene where Bayard helps Alice.

So, as a summary - Bayard knows that he is being used and helps Mad Hatter in this scene leaving Alice.