You could always rent a 4x4 truck in Corpus or in your hometown and take it to the park, or carpool with a friend who has a 4x4. There's no rule that says you must OWN the 4x4 vehicle you take down-island.
For that matter, there's plenty of beach access before the 4x4-only area. And bicycles aren't banned.... it'd be a challenging workout depending on the sand conditions, but you could haul a goodly amount of stuff with the properly equipped bicycle. (I'm not being sarcastic; it's an option if the beach conditions are just right--you want some harder packed sand vs. the light and loose and deep stuff that happens after a string of hot, dry and or windy low humidity weather.)
Just think about handicapped folks and others who cannot enjoy our parks as easily as we can, if at all. I understand you'd prefer to use an
ATV, but don't focus on the negative and instead try to look at all your options and make your trip a reality. We are blessed to have such a beautiful, thriving (fishing, sea turtles, shorebirds), undeveloped coastline in our state. It may not always be that way.