Although I've totally resolved the question which began this thread, I'm still struggling with the rest of lemma 3.
For the cases where either v1 is not a or vn is not b, I understand completely that such circles described in the paper would resolve the issue. I also understand that we have the right to move the circle small distances. The problem is that the circle may be touching the polygon outside of gamma(A). Without appealing to geometric intuition, it's hard for me to be sure that small transformations of the circle won't cross tangent lines outside gamma(A). If I draw pictures, I can be sure that this kind of problem won't happen. But, if you appeal to geometric intuition, you might as well just accept JCT without proof.
I'm fairly confident about the case where v1 is not a and vn is not b. However, the case where v1 = a but vn is not b, still troubles me.
I really hope that once I've mastered lemma 3, the rest of the proof is easier (for me). I've wanted to learn a proof of this theorem for literally about half my life, but I've always given up my earlier attempts.