Considering how CWRU got to running three HS
comps each year...
If those dates you suggested
conflict with other competitions going on in the area as
you suggest, then you should defer to the fall
semester so you can all think out what you want to do and
take that extra time to make contact with other
schools.
The other thing I suggest is to determine what those
other competitions are and go help those tournaments
out. Make contacts there and (if you're that
organized) announce a date for your competition. But
obviously, make sure the date you choose doesn't conflict
with any other established competitions or important
dates on the academic calendar (yours or HS).