People are doing this already! But I would go to one if it where close to my home or work!
My wife makes an amazing grilled cheese and we talked about a grilled cheese truck before.
Not an original idea though, I have seen it on the food network for the last couple years.
http://foodtruckfiesta.com/grilled-cheese-food-truck/
http://www.facebook.com/roxysgrilledcheese
http://thegrilledcheesetruck.com/
http://www.chedds.com/
http://meltbarandgrilled.com/
http://theamericansf.com/
http://grilledcheesegrill.com/
Also $2-$3 sounds like a great deal I believe it is unrealistic as a price point, you have overhead costs for the building, utilities, staff, equipment, insurance, and the cost of food. I fear the cost of good quality bread, good meat and good cheese will put you over that price point before the other factors are even considered.