Strictly from memory
Burning Hills did host the palmetto Classic and Kohls Cup for certain ages
CFC backed out of the process because it did not have enough fields (please note past tense - do not know why)
CESA offered MESA but without volunteers as it drains the volunteerism for in club tournaments
To my knowledge the Challenge cup is not a problem...i.e. the number of teams are finite - 32 in the fall - 16 in the spring.
The problem is more with the Classic Cup and Kohls cup. To the best of my knowledge this is open entry, therefore you may not know how many teams will choose to enter, thus the number of fields issue.
Now that there is a regional cup for classic play (I think there is a national as well) The classic cup should also be limited to 8 teams and have the same process as the challenge cup (i.e playof game and then state cup).
Field needs will be reduced considerably.