You mentioned floating the barrel, but have you glass bedded the action? I had a Savage with a thin barrel in .22-250. The first two shots in a five shot group would often touch, then the group went all to hell, especially with the last two shots. Sporter barrels heat up very fast, and you can kiss acuracy goodbye. I always reach around and grab the barrel about an inch or two in front of the action. If it's too hot to leave my fingers there it's too hot to shoot. Lightweight barrels don't usually do too well at the bench in the best of times. Drink a coke, have a smoke, go for a whizz, anything but let that barrel cool. That or, do what I did. Put a Hart SS 26" varmint barrel on it. My .22-250 now groups in the mid .2"s. It wasn't cheap, but I smile a lot more now. Best wishes.
Cal - Montreal