If market is more than the ED makes, and you want to make anywhere near market, you have to leave. They're dramatically underpaying market, and have to for structural reasons, so they won't be able to hire someone anywhere near as good as you. To retain you at market, they'd essentially have to go on a special fundraising binge to pay your salary. That's untenable in this fundraising environment.
Also, the title thing is hard because you're doing two very different jobs: software engineer and helpdesk.
I'd recommend that you leave if the money is important, but you should (if you reasonably can) try to lock things down so that there won't be much in the way of maintenance of the website. The office computers and phone can be troubleshot with one-off contractors (again, helpdesk); there's no way to "bring in someone from the bullpen" to put out fires on a complex RoR app. (For one, NPs have no idea how to recruit or verify the credentials of someone like that.)