Here is a thought or two.....
Sew the lattice back to the block with a gentle concave arc to match the convex arc of the block sides. You can still do this by machine and I think it would be more in preservation of the original design. (I do wish I could see what the original quilt looked like as far as the layout goes, perhaps you can find another image like it?) I do not think the sewing will be too much of a challenge compared to the deeper arcs of say Robbing Peter to Pay Paul or Lemon Peel blocks. In the end the lattices will be slightly concave, but the rows should lay flat.
Practice first...
Start with a piece of mylar/template plastic and draw the outline of the blocks arc. then add seam allowance to the inside of the arc. this should be the cutting line of the edge of the striped lattice work. Don't cut it however until you have tried the idea out and know you want to follow through with it! In fact, i'd use a piece of scrap fabric to try it out. Just make a stripe lattice the same size as the ones you have and a solid muslin or denim the size of your blocks, play with them and see if the technique is to your liking. If not, no harm done!
Good Luck! Cathy vJ in MA