Corn GLUTEN meal will cost you a fortune for an acre.
Corn meal will not do anything against them.
Stickers are an annual grass so you can work against them in the winter. You can get excellent results against the burrs simply by growing great grass for a couple of years. If you want something to really shade out the sticker seeds in the spring, you might check around and see if tall fescue grows out there. It should. St Augustine was my first choice but you said your wife wanted a finer bladed grass.
If you are willing to really go off the deep end, I would suggest you plant both tall fescue in the spring, but before that, this winter plant some white clover seed. If you get even coverage with clover, rather than the clumps we are all familiar with, a clover grass mix is beautiful at all times. Then mow it every few weeks at the highest setting your mower will go. With this mix, you can easily get oversized wheels and mow up above 4 inches. Be sure you inoculate your clover seed with bacteria as directed on the side of the package the night before you plant. If you do this, you will never need nitrogen fertilizer again.