Well they are not asking for which one is best, they are asking which would you choose and give arguments as to why you would go with that. So you could pick a few of them for various parts of the system.
Generally in a situation like this Rapid Application Development or Prototyping is a no brainer because the system is going to be used heavily by people and everything in the system is entered, manipulated by people working on "the front lines". Having said that, providing them with quick demo projects to feel out what works and what doesn't would be a great way to nail down the specifics. The client will have to play with some of the demos and try out the software workflow to see if it matches what they do as a company because efficiency is key (they want to process people in this system and the quicker and more efficient they can do that, the better. So workflow is key)
For the same reasons above I would also go with an incremental process. Demo something, let them try it, get their feedback, scrap what won't work, keep what will and then use that as the base to build bit by bit on it. That way you can keep getting the functionality out there and nothing will stop the work from continuing. You can have your team developing the features of the system that was last approved by the client while you are meeting with the client to get the next iteration of the project in place. It would be better than starting and stopping to collect information since you want the system on a 3 month time scale (which I think is pretty aggressive for a project like this).
But that is how I would tackle it.