I go to a non ABA accredited school in Mass. We can sit in several states after passing the bar here, some after experience and some immediately.
I know our school has been active in going to various states to make the case for allowing our graduates to sit for the bar, which has helped open some doors. It took alot of work by the school and a few court actions (including suing the ABA,) but they were successful in some states. You might want to push your school and its alumni to lobby the nearby states change their rules and allow you to sit.
Remember, each state has the right to set its own rules. The states that only allow ABA accredited graduates to sit are lazy and/or have buckled to the ABA's pressure by simply defering to the ABA. If non ABA accredited graduates could sit, the ABA schools would have a tougher time getting people to pay 30K a year. The states should go out and do their own homework to see if graduates of your school are qualified (which I bet most are) instead of giving a blank check to the ABA to decide which schools qualify and which don't.