Annulments have certain requirements (especially an existing situation at the time of the marriage) that aren’t easy to meet; when the spouse fights the annulment (as Ted Kennedy’s wife did) then it is even more difficult.
I suspect the divorced/remarried Catholics that want the ban lifted have already gone this route unsuccessfully.
A very good friend of mine got an annulment. He told me if you got the $$$, you basically buy it and it doesn't take years of waiting.