This is a super cool block. I think the lozenges (elongated hexies?) could be created using 3 squares and the 'flippy corners' method. You would have to trim the excess triangles. I am not sure what to do about the ends of the bowtie. Those pieces would be a challenge to cut, if not to piece.
They might be done with the "3D" method, which uses all squares. It is basically a 4-patch with an addition. But the interesting part is that one bowtie block becomes the corner of the next one. That might be a bit fiddlier to piece.
Just what I was thinking!
It's certainly an interesting puzzle