I had migrated to an anchored putter for the reason you posted. Then the rules changed and the group I play with (we always have a friendly bet and team game) play by the rules so I went back to experimenting. After trying many methods and putters, those I already had or borrowed, I went to a mid size putter I customized to 79 degrees upright, 41" length and a 400 gram head. This is not a belly putter, it is a long putter lie angle set up at belly putter length and head weight. I use it with split hands and putt in the same manner as if it were anchored, but using the top hand anchored by elbow extension and not anchored to my body and lower arm to swing the putter keeping the shoulders still. My group watched me for a while to see if I was violating any of the rules.
It took me a bit of practice, but I am putting as well right now as I can remember. The shorter, 10' - 12' putts and in, immediately improved. This was the primary reason I went to the anchored putter over 10 years ago. The longer 12' - 25' putters took a bit of practice, but I am better now in this range than I was with the longer anchored putter. The 25' plus length requires more time on the practice green when the speed varies from the previous time I played. I have to putt enough to establish how far to take the putter back in relationship to the length or on slow greens I will practice a hockey style use of my putter.
Many people struggle with the distance of a long putter, 39" plus, because they alter their stroke. I have watched this time and time again with people who told me they could not effectively control the distance of the long putter for long putts. There are two solutions, one don't change your stroke, keep the anchored hand in place. The second is use a different stroke you practice for the longer putt. Hold the putter fixed with the split hands and putt with your shoulders - hockey style. Some use this stroke all the time (Bernhard Langer) with a long putter. Both work, but you have to decide and then be consistent. You will find it difficult to be consistent with the long putter if you mix the elements of the two methods - that is using your shoulders while also using a lower arm swing or introducing your hands to get more pop into a shoulder stroke. You can use the fixed hands consistently on the really long putts if you practice.
Personally, I seem to have a limit around 45' on medium greens of good distance control with an upper hand anchored approach. It may be one of trust. I have seen others always use this stroke regardless of the length of putt. For me I tend to want to help with my body with the longer distance and it usually produces poor results. I will pace off the longer putts and when I exceed 45' on a flat putt I go to the fixed hands - hockey style - and use a shoulder stroke. Why don't I do what Bernhard does, use it all the time? I am consistently better with the other stroke inside of 30'. In the 30' - 45' range I just go with what feels best at the time. For me this is probably more about eye sight more than method.