Great to hear and I understand about the mat size - I have the 18" and have an extra 5" internally so I assumed (yes that gets me into trouble!) that you would have 5 extra inches as well in which case it would work. Bummer.
The two outer wheels should hold the mat in place. Are you sure the pinch wheels are positioned over the grit shaft Depending on what you are cutting, that should be firm enough to hold it. So do double check the positioning of the pinch wheels. Also, if the pinch wheels get "gunky", they will not hold as well.... So please check both. That should be helpful.
Sounds like you have a huge project going - wow. Hope you will share with us when you are done! Eager to see it!