Unfortunately, you are s*** out of luck there - weather conditions are random in career mode and cannot be altered at all.
With that being said, you should really upgrade your steering - buy a gamepad! They're not that expensive and can deliver decent results. Wheels obviously are the even better option, but they're much more expensive - at least the good ones. If I were you, I'd steer clear (pun intended) of the cheap ones, like the Thrustmaster Ferrari Red Legend Edition. It contains a rubber band which will wear out in no time at all, and the screw/lever system with which you can attach the wheel to the table wears out pretty quickly, too. After having been thoroughly disappointed by that one, I switched back to my trusty Logitech Gamepad F310, using the analogue sticks for steering and throttle/brakes, which works astonishingly well, even in heavy rain. I don't think I'll buy a wheel again any time soon ...