what do you mean with base position and the beginning, could you post a screenshot and I will have a crack at it?
If you mean, you want the stabilizing effect while being able to aim the barrel, I think there are a few ways of doing that, but what kind of balance do you want? because you could first use animation blends, like aim offsets, or you could use a complicated branch system with a lot of conditions, or you could change the base input values based on your aiming, what exactly do you want to achieve?