I believe the hours listed are the hours when lifeguards are present. Outside of those hours, the beach is open, patrolled occasionally by FIBP, but they won't bother you unless you are doing something clearly illegal (fireworks, un-permitted bonfire, kegger, etc.) And of course, solo ocean swimming when no lifeguards are present is never a good idea.
I dont think it will be a problem. You have to pay to park since it's a state park, but I don't remember seeing any gate or anything there. I think you should be fine to go there at 6am.
It seems pretty chill to go down before life guards and such. I went down to watch sunrise often, and came down at night. The sand sweeper might get ya, but not on the North side.
Dawn patrol is doable this time of year.
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