If I am understanding correctly the images and details, I think it may be because there is too much resistance on the fishing line. From the point where the line is attached inside the head does it have to angel to the point where it comes out of the head?  Also then when it goes through the hat is there any angle?  For a marionette you really want the string to come straight up and out of the head and hat without it having to rug against any surface.

With mouth movement it can be done two ways you can have the string open the mouth when pulled (how you have it rigged now) or you can reverse that and have the mouth open when you release the string.  From your first image it looks like you may be able to change and use the second method. The weight of the jaw would make it naturally stay open. You would attach the fishing line to the lower jaw where I am seeing what looks like a piece of string elastic. That might give you a straighter shot up through the head.  Part of the issue is that with a skeleton the hinge point of the mouth is so far back on the head.  In a regular set up this hinge point is more centered in the head which can allow for less resistance on the path the line must take.