I don't own a trailer, so consider that piece in my comments.
Assuming, you are committed to cycling to that store and not willing to cave to the car pressure, and considering the second line of your sig, "Competitors work until they get it right, but champions work until they can't get it wrong", I believe the champion grocery hauler would not be a kiddie trailer, but one built specifically for cargo.
Kiddies have round bottoms, don't weigh as much as a month's worth of groceries, and are bored after an hour or so. Thus, kiddie trailers are designed for short-term carriage of round, soft, lightweight things. I've never seen one rated for more than 75 pounds. The fabric floors keep soft, round bottoms comfy, and they have things your don't really need like a roof, sides and screened windows. Essentially, it's a tent strapped to a frame.
Take a look at cargo trailers instead. The gold standard is
BikesAtWork.com. Even their smallest model is rated for 300 pounds, well beyond your anticipated need. If you look through the site, you'll see they're designed to work well with those large Rubbermaid totes. Conveniently, camping coolers are about the same size.
I'm guessing you could get by with the model 32A and four totes or coolers. But for only a small amount more you can get a trailer twice as long. For a long ride in Texas heat and wind, I'd think about a longer trailer since one layer of totes will have less wind resistance than two.
You're talking about some serious hauling in some serious conditions over a considerable distance. A cargo trailer will help make it so that you can't get it wrong.