My singing voice is quite good, I can hold a tune etc., but I always used to shy away from singing backing vocals whilst playing bass for all the reasons mentioned above - countermelodies, insufficient brain power, whatever.
I've mostly got around this by trying to learn new songs for my covers band all at once, learning to sing the backing vocal at the same time as getting to grips with the bass pattern. Sometimes this doesn't work, especially if the bass pattern is particularly busy, but the approach certainly got me started singing bv.