I think if you space out your priorities a bit more, right associative operators could be handled by adding a bonus to their weight:

As written:
String:  3 ** 2 ** 4
Weights: 2  9 2  9 2
[download]

Now change the algorith as follows:

• Multiply normal weights by some factor, say 10.
• Multiply level bonus by the same value.
• Add 1 to priority for each occurrence of a right associative operator.
• Swap priorities of parens and POW.

String:   3 **  2 **  4
Weights: 20 80 20 81 20

More complex string:
String:   3 **  (   6   -   1   *   4  ) **  2
Weights: 20 80 90 120 140 120 170 120 90 81 20

String:   3 **  (   6   -   4  ) **  2
Weights: 20 80 90 120 140 120 90 81 20

String:   3 **  (   2  ) **  2
Weights: 20 80 90 120 90 81 20

String:   3 **  2 **  2
Weights: 20 80 20 81 20

String:   3 **  4 
Weights: 20 80 20 

String:  81 
Weights: 20
[download]

Obviously, the weakness of this approach is that if you have too many right associative operators at a level, your priorities collide. You can make "too many" arbitrarily large by using a large multiplier for the original priority scheme.

I'll leave actual implementation up to you.