In Texas bobcats are considered a varmint and as long as you aren't sellling the pelt you can shoot them anytime. LOTS of bobcats in Texas. Their mating season is in February.
Not sure why bobcats have such a mystique about them, they are pretty comparable to a coyote in predation, maybe even a little more capable.
P.S. - For some reason I like bobcats over a coyote as well. I won't shoot a bobcat with my hunting rifle or during the summer or early fall because if I take one I want to keep it's pelt.
Here's my Christmas present from Santa last year.
He sits with me in my office now.
I've shot or trapped almost a dozen coyotes and never had the desire for one of them to sit in my office with me!