I have a Bernina frame and have been using my Bernina 440. I have quilted 3 bed size quilts - full and king - but it's a challenge. First of all, I always use Warm and Natural batting which is thin. You could also use flannel. Also make sure you are using pins to attach the backing to the leader. Anything that is thick, like Red Snappers, takes up valuable space. Make sure you roll the quilt tightly on the take up rail. As you go the quilting area gets smaller so plan an easy quilting pattern, like a meander.
Fortunately Santa brought me a Bernina 710 which has a 10" throat and I am looking forward to using it on my frame in the near future.
You may want to keep your eyes open for a gently used machine that you can use just for quilting on your frame. Some dealers offer good prices on reconditioned mid and long arms.