I'm using google maps to look at the link I mentioned above... Indian Mound Park might work.

It looks like there might be a spot just East of the Manasota Beach Public Boat Ramp.

The North Jetty Park Primitive Boat Ramp looks real promising. Easy access to the Gulf.

Maybe the Turtle Beach Public Boat Ramp? You might be able to park on the East side of Blind Pass Rd and hustle your boat to the west side, then launch from the beach.

There is a small beach just south of the Sarasota Sailing Squadron.

Maybe Centennial Park?

All this is guessing from the information that the fwc provides in the link I posted earlier. Let us know if any of these pan out.