Justin,
It sounds like you have a pretty clear idea of your obstacles in setting up an alumni association for your institution. I believe you also see the benefits. As you mentioned, your challenge is to show the Board of Directors the benefits and help them understand the issue of cost.
But, the bottom line is a RELATIONSHIP - we want to continue our relationship with our graduates and we want our alumni to continue a relationship with us. It can be a mutually beneficial relationship. That's the issue your board is going to have to agree on. The cost/benefit can be somewhat blurred, but the relationship can prove to be very powerful for your institution.
Good luck!
June