I think that your view master model is a good direction. Your shutter could be a slit that exposes the film as it rotates when you press the trigger.
Or the film cartridge could be fixed and the trigger rotates a disk with the slit. This might be interesting in that each shot would have a slightly different perspective.
You are going to have to correlate the width of the slit with the speed of advance by the trigger and the focal length of camera for a proper exposure.
Very similar to this http://www.noblexcanada.com/noblexproducts.htm
For example, at f16 on a sunny day what radial speed across the slit equals a 1/800 th of a second shutter speed ?
What are thinking as far as a lens ? I do not think that a pinhole will work for your lens in that the trigger mechanism will not yield a slow enough shutter / slit speed.
I'm thinking that you could butcher a holga to prototype your 3 shot film cartridge and trigger.
Do you know how to develop film ? In order to test your prototypes for proper exposure you or someone else will need to. You could prototype in house with black and white, then when the variables seem to have been worked out switch to color.
To complete the view master model maybe you will use E6 film, returned to the user mounted similar to the view master slides, perhaps projected onto a wall by a simple cardboard projector as well.
There are certainly many here that can help you in that regard.