Supporting hex grids is a huge task. It requires having 2 separate TileSet resources for the parent and child. But I've got ideas.

I manually cropped and assembled this tileset from https://youtu.be/Uxeo9c-PX-w?t=428 (the same frame used in the readme) which I hope could help develop triangular and hex grids further. It's a bit complicated since it's a "tall" tileset, so I'll be looking for a simpler one for now.

Each triangle sprite is 64x83, and when pieced together they become 85x72 hex tiles. I don't know how the math works yet.
Unfortunately, Godot does not support triangular grids. Fortunately, they're just square grids with extra steps.
I think the geometry of the display_tilemap can be computed as follows:

display_tilemap.tile_set.tile_shape = TileSet.TileShape.TILE_SHAPE_SQUARE
display_tilemap.tile_set.tile_size = Vector2(self.tile_set.tile_size) * Vector2(1, 0.5)
display_tilemap.position = i_dont_know_yet()

But you'd need display_tilemap to have a different tile_set for this to work. It can't reuse the parent's.