Auburn is not on the county's list of approved private placements, so you will not get funding.
Eligibility is good for three years, so if you return to public school before eligibility expires, you won't have to start over again in terms of qualifying. Goals and services, including placement, are only good for one year, so you would have to renegotiate that again. As a practical matter, even if you moved her to the asperger's elementary school program, you would have to renegotiate when she moves to middle school anyway; would you send her to auburn for so few years that she would still be in elementary school when you wanted to return to public school?