It's valuable having an enthusiastic early adopter, so I would want to use your situation:
To do just enough to put that first reference client on a viable footing for the next 12 months
To use the learning to construct your pricing proposition for new clients
I like keeping things simple. So in your situation, I would work out exactly what I need to make from client #1. Whatever the figure comes out at, I would go and engage the client and look to sell in what would be a sweetheart deal for them in exchange for the benefit to you of having a lead customer who is willing to continue to partner with you as you develop the product. And that could include exploring how to price in a way that works for similar institutions.
If your client loves the product, there should be ample scope to come to an agreement that gives you both what you need.
But what if any discussion about them paying for the product hits up against your "free" competitor?
Well, that's a great test of your core value proposition. You have all the aces: a running trial with enthusiastic users, a product that does what they need, a combination of technology and relationship. If your client really won't pay you a cent for that, you're going to have to have a major rethink of your business model!
As for pricing as you engage new clients, you have a blank canvas. Pricing per user (or, similarly, user block pricing) is a possibility, regardless of how your system is deployed. Or it could be a consulting sale - charging for the system, for time and materials for customisations and installation, and ongoing support and upgrades. Or... Or...
How do you decide? Well, the structure of pricing your product needs to map both onto the way your target market perceives its value, and the way their buying processes work. You should be looking at other third party software used by the clients you are already engaging with and finding out how they came to buy it in the first place and how they pay. And you should be looking for emerging trends, so you can line up with what the switched-on faculty is looking for.
Capturing what learning you already have in this area, and engaging with clients to close some of the information gaps, should be your priority. You're not in the situation of most web start-ups who can experiment with pricing structures and levels. Ideally, this would be something you would have worked on in parallel with other activities to date. If you haven't, you should expect it's going to take you some weeks to have pricing that you're comfortable optimizes the opportunity going forward.