This is a really great suggestion - many non-profits will go for it; keeps their staff (you) happy, keeps you connected to them, and they can bother you on "freelance days" too.
But that's the problem: you've probably already got more on your plate than a time-serving big-corporate IT person would. And being present means that you're first and foremost available to the people in your vicinity.
Keeping the relationship with the NP in good shape can be hard work; some colleagues may consider you a money-hungry sell-out (and the conversations with your customers may just reinforce this idea), more problematically keeping the perception of balance in your interest in the NP versus outside clients could be difficult.
Good luck!