Hi I went down lake turkana (north to south) last year on motorbike, pretty hard. if you have a 4wd then I think it would be fine so long as it is dry, what season is it? I have heared that it is not possable in the rainy season, what will it be? for you it will be, first bad roads, then rocks then more rocks and then sand. then more rocks then sand, more sand then Eithoipa! just joking its not that bad, certainly not in a 4x4 generally you are driving/riding in ruts all day. The tribes in the area are amazing! There are however bandits in the area, however they are generally more intrested in cattle in than travelers.
I would also recomend G.P.S. There are detailed maps avaliable at the park gates, might be a good idea of picking one up as there are quite a few tracks in the area. Have a great time!