I work quite a bit with healthcare and also have clients who actively sell software into the healthcare industry both to carriers, providers, and even government orgs. I would be inclined to provide them contract options with three and six month exclusivity terms, and then even provide a long term rate that makes it worth your while to maintain exclusivity in that territory with specific terms you deem reasonable so you can push the product elsewhere. You may request in turn for the exclusivity period, different terms so it is worth your while to provide the exclusivity. Perhaps they provide reference to any other sales opportunities outside the territory, pay increased rates per longer term they request the exclusivity, agree to let you use their band in advertising in the future pending they stay a customer, etc.

This is just a negotiation point, they have come out in want of exclusivity, figure out exactly what it is what you want out of the opportunity and start the negotiating points there.
Good luck!