You could definitely do cross hatching. You need either a walking foot or a dual feed foot. You want to set your stitch length to 3mm.
If I were quilting it, instead of cross hatching, I'd probably stitch the lines so that each little section has parallel lines going different directions, and maybe the bigger sections I'd split into 2 or 3 subsections & then follow that same design of parallel lines alternating directions by section.
Really adorable quilt! Can't wait to see it quilted!