igbal,
I could be mistaken, as I don't live in Texas, but I think even private landowners have to follow the seasons set by the Texas state department of Fish and Game. Exotics may be different, but my best guess about the deer would be they could only be hunted during regular deer season.

I'm not sure how the law works in Texas, but sometimes farming landowners that have a problem with deer destroying their crops can get special permits to cull the deer herds on their properties. I don't know if that is year around or only at certain times. I don't know if they could enlist any help with those activities either. Something that might be worth looking at though.