Said this before on this site, but everyone should realize "no surfing areas" and "surfing areas" are in a surfer's best interest. If you were to hit a swimmer with your board and injure them, you would be liable for any damages.
I will argue, along with any surfer, that surfers deserve many more surfing areas, and there should be discretion on where these areas are based on what sandbars are working.
But anyone that criticizes the "no-surfing beaches" just because "they suck" is a fool.
With all that said, you should have a decent argument if you are using a beater board (without fins) in a no surfing zone. Whether you stand up on it, drop-knee, or lay on it shouldn't matter. But because no surfing zones are to protect swimmer's health and the surfer's liability, using foam, finless board should not be an issue.