This topic has come up on one of the threads (costume?) but I can't remember which.
The gist of it was about the moisture wicking material, especially the version the rD shirts are made from, is a total PITA to work with. One individual mentioned making a quilt with theirs and I think they mentioned the PITA factor as well.
The stuff really squirms around a lot. I lop off the sleeves of my race shirts to make them a short sleeved shirt and the most successful approach I've had is to fold the material over for a straight hem with heat-activated fabric tape underneath and the material held in place with a combination of pins and/or painters tape. Iron the hem to get the fabric glued back on itself and then go to the sewing machine for a a stitched hem.
So if you manage to cut out the shirt fronts and get them square (your first challenge) maybe try heat tape before trying to sew it to a quilt. And now that I type it out, I'm wondering if a better solution might be to use adhesive sheets and iron the shirt front (at least the part with the graphics) to a cloth substrate that would make it a lot easier to then cut the shirt front out to a (presumably) square shape and then sew it all to the quilt.
YMMV