Hi mkultral25,
I am a primarily a matchmover in the VFX business and have worked with tons of green and bluescreen shots on feature films.
Without any markers, it will be very difficult to get a good camera in Maya (after a camera track in matchmoving software). First the shots will need to be tracked, no doubt. Because of the camera movements, markers of some sort need to be placed on the walls (as many walls as possible) and on the ground/floor for parallax so whichever tracking software can solve it. Without an even array of markers throughout the environment, the solve will not be good.
See if the greenscreen surface can accept tape. You know, like those circular stickers that can be found at any office supply store. Or use white tape and make cross markers. Ideally place as many at exact distances from one another and record those distances (you know in the X and Y directions). These measured markers are better to create a "survey" set of data that can be plugged into the tracking software and checked in Maya once its exported from the tracking software.
Manually tracking just using Maya will be very difficult due to the complexity of the shots as you describe them. Not impossible but will be more time consuming than you will expect.
Hope this helped. Let me know if you have more questions. Good luck!
Do you have access to any tracking software?