I would do this: I-39S > I-55S (Bloomington IL) I-55S > I-40W (W Memphis AR) I-40W > I-30W (Little Rock AR) I-30W > I-20W (Ft Worth TX) I-20W > I-10W (West Texas) I-10W > I-405N (Los Angeles CA) I-405N > your Sherman Oaks Exit Please note - Sherman Oaks is in a valley so you will have to drive over the Sepulveda Pass to get there. Technically it's a mountain, but there is a very low chance of snow due to perfect southern CA weather. I-405 is a major commuter route in Los Angeles so it's an easy traverse. Safe travels!
We drove from Los Angeles, Ca to Dearborn, Mi during February of this year and also wanted to avoid the mountains due to snow and other winter conditions. So, going from Wisconsin I'd head South and go thru Missouri to Oklahoma. Head West and go through New Mexico, Texas and, Northern Arizona. In between Alberquerque and Flagstaff you can check out the Petrified Forest and the Meteor Crater just east of Flagstaff. Take the 40 from Flagstaff to L.A. If it is too cold go further south to Phoenix, Az then, take the 10 west to L.A. The mountain passes into L.A. are the easiest you'll find. Enjoy your stay in Sherman Oaks.