First things first. I don't think project management approach or software development lifecycle of your choice is your problem at the moment. What you have here is more an issue with leadership and/or team management.
I know such setup to be very difficult to work out. At the beginning everyone has a lot of energy to pursue the new thing but after some time most of people start considering it as just another job on the top of their main one and their private lives. Then the question appears: what's in it for me?
Now, if the answer for the question is money and money only you may be in serious trouble. I don't know what your timeline is, but working part-time on a regular basis and as an addition to your regular job will likely burn people out in a matter of a few months. Additionally, if they don't see results they're for in the project (money is supposed to be at the end) they may just decide to cut their loses (invested work) and abandon the project.
Much more sustainable motivation, and motivation you should look for, is when people genuinely want to create the product because they strongly believe in it or they're having fun with building it for technology reasons or whatever else. But then, such motivation wears out as well -- people can become tried waiting for the first version to go live or they decide to pursue another technology or they just don't want to sacrifice their private lives anymore.
Without answering the question why people are in the project in the first place you won't find an answer to deal with their decreasing motivation.
Once you know that, as a PM of the project, you can try to adjust the project environment in a way which would be more motivating for the people. A few ideas:
Discuss engagement issues openly with everyone. Try to find root causes. Try to learn what can be changed to improve people motivation. Be honest and open with what you can offer. Leave open doors for those who want to leave. It's better to have fewer engaged people than a bunch of crappy contributors.
For those who are in the project for the money think of partial payments so they actually see what they get for their work. It's a bit tricky since if people decided to go, they'd take the money and go anyway.
For those who are just tired think about smaller contributions they can make so they don't sacrifice as much of their lives as they currently do.
For those who care about technology try to give them enough freedom in this area within borders of the project. Sometimes it doesn't really matter what kind of framework you choose to build something but it can be a delighter for developers.
Look for potential substitutions. As harsh as it sounds people would eventually abandon the project as this setup is really hard to deal with from everyone's personal perspective (think 8 hours of daily job and then 4 hours working on the project). If you think about new people in the project focus on those who would be there for non-monetary reasons.
This why the project is funded that way. From what you write I read there could be 3 people working full time on the project plus yourself. And then for some reasons it isn't build by people working full-time or contractors but as a side-project for everyone in the team. The question is why? Do sponsors really believe in the success or do just want to try it out and do it cheaply? If the latter is true, don't expect much of motivation from the project team. If sponsors don't believe in the project why should the team?
I know there are many questions here instead of answers but there isn't some kind of magic wand which would solve such a difficult situation. When it comes to people motivation you have to do the grunt work first, otherwise you won't be successful.
