You could use Agile methodology here but obviously you'll have to adopt it to suit your needs.
For example if you don't have enough access to the real customers that someone with the best understanding of your goals will have to act as a customer proxy. I would also suggest trying to get more access to the customers - almost everyone try to appear more busy then they are and there is usually a way to resolve that obstacle.
Make sure that the limited work time your team have they have at the same time. There could be no Agile approach when you could not work together.
You could definitely use story-based estimations, iterative development process etc.
What is really important is too give every team member a clear and unambiguous understanding of how the Agile process works and what is each person's role in the project. It's very easy to say that you will use SCRUM but unfortunately without real understanding and experience that will not really mean much.
Some advice:
- Educate your team members
- Get a list of what you would like to deliver if you would not be limited by the time/resources.
- Find out what is realistic to deliver given your constraints. That will probably be not much. Don't try to be overly optimistic. Focus on what you could really achieve.
- Make sure that your real customers are on board for that.
- Use short iterations (1 week or less). Make sure you could deliver fully tested product by the end of each iteration.
- Show your work early.