I would put it on the longarm. The stretching provided by the rollers will help to even out the wonkies, but I would try the pressing first, maybe with starch to help get everything back into shape. Quilter's Dream makes a lovely poly that has lots of loft that will help with the puckers. Because the sashing isn't straight, you really need to do an all-over quilting job on this quilt. The Baptist Fan would be lovely, so would swirly circles. I have quilted some very interesting quilts on my longarm (yeah, they are mine :oops: ) and they always turn out looking so much better after they are quilted!!! Please show us the end result!
If you decide you just have to quilt it by hand, use your LA to do the basting of the layers - so much easier!