Finding an appropriate business model is situational and may evolve as you engage with the user community. You could start by working in public with an open source license [1]. Your initial focus is customer discovery and building a community of stakeholders willing to invest their time and accepting the adoption risk. Once you have advocates, they may have suggestions for an appropriate business model. For example, the existing non-profit may decide to apply for grants, keeping the work open source and acting as the fiscal sponsor for the project via open collective [2] supporting a paid position. In my experience, getting caught up on recapturing early investment, being concerned about secrecy / proprietary value, or being worried about a competitor are often not the best use of mental energy early on. Conversely, sharing more about your space and ideas may win you collaborators and potential customers you didn't anticipate. Alas, none of this advice may apply, the right approach is specific to a particular opportunity. Good luck & have fun.
[1] https://press.stripe.com/working-in-public [2] https://opencollective.com/become-a-host