This seems like a huge uphill battle.
Lax is a spring sport in school and I believe most people are of the mind set that Club lacrosse should not interfere. If I am correct in that assumption than the league can be in the Fall or Summer.
I think the Fall is out due to Football and Soccer.
I think the Summer is out because the Tournament circuit is too well established. Especially for the HS player because most are geared to be Showcase for college recruiting.
I don't know how soccer developed, but it seems the economic structure of Lacrosse on LI has usurped the ability to go to a league. I also think the current college recruiting "pathway" will make people leery of changing to a league format.