Maxxis Bighorns are an all around good tire i have them on a 2007 can am outlander 650 maxx xt and they work great in most of the terrains we use the quad in mud, loose dirt, loose gravel, packed trails, in snowbanks.
Most likely it is the same switch located in the same place as in earlier models. There are Big Bear 400 manuals in the download section .. Check them out for the placement of your switch .. It may be something as simple as a bad ground .
I would definitely go with and all terrain tire. I use them on my hunting quad because i go through so many different types of ground including flat ground. As @06kfx440 said a mud tire on smooth ground makes for a bumpy ride.
Ebay has a ton of all terrain tires.