You can do it if you have a very gentle touch, but no matter what you are going to break some roots. If you are gentle from start to finish there would be no reason to break up your roots unless it is severely root bound. If you are root bound, and can't even get your fingers in the middle of the root ball, just grab a knife and make one cut from the bottom, going to inches up through the root ball, just two inches. you can open it up a little if you like, but the roots will work themselves out. Severing the roots causes hundreds of new feeder roots to shoot out from the cut.
If you want a stress free transplant, put one of your existing buckets (I'm assuming you are starting with 1gal) into your new buckets (I'm assuming you are going to 5gal) with a little bit of dirt already on the bottom of the 5 gal. Now fill in dirt around your 1 gal container until the bucket is full. Slip out your 1 gal container and you have the perfect fit for your transplant, I do this, and I usually have tap roots sticking out of drainage holes in 5 days.
Weather or not a plant is root bound, unless it is flowering, the roots will find new dirt on their own, probably with less stress and time than if you tried to assist them.