Helps a little, but the issue you have is that the head is not heavy enough if you are going to add more weight to the grip end. 125 gram putter grip is “counterbalancing” to a certain level. I think the only way you still have any head feel with the 125 gram grip on a 360 gram head is because the putter is 38″ long. The process is truly a trial and error process based solely on what feels best for you. The technical side of why to do it is basically this: heavier putter heads (380 grams – 450 grams generally) are more stable in motion and have a higher MOI (resistance to twisting). That being said, heavier heads with a standard grip can be harder to start in motion and for many harder to use consistently, especially on longer putts. Heavier heads with standard 50 gram grips usually feel too heavy for most players anyway. By adding weight on the grip end, you are “counterbalancing”,making the head feel lighter, but also making the putter more manageable and easier to start in motion. Much research has been done to show how a heavier head with counterbalance affects the ability to stroke a putt more consistently and majority of the results points to the positive. I believe you will need to either add weight to the head you have or get another putter that weighs at least 380 grams. The amount that you use to counter balance is difficult to say. You will need to see how a 125 grip with the 380 gram feels and performs. Maybe you need more than the 125 gram grip to give you feel you want. You may need to try several weights before dialing in the best combination.
I wish I could be more specific, but that is the only way to find the right combination for you.