I have the same challenge as the OP, and agree with everyone's advice: learn the bass part 'til you have it down cold, first. Frankly, I have the same challenge with backing vox on some songs, too, and my solution/advice is to nail the vox when it's lead, and flip a coin on backing situations. What's more important for the particular song, the backing vocal line or the bass line?
I think the issue comes from this: there are very few bands with a lead vox/bassist. Let me rephrase that a bit: that band set up is a minority. So, you're tackling parts written by two different people who aren't really writing for the other person.
When I played in hair bands, it wasn't a big issue. Poison, Dokken, AC/DC, even Judas Priest - the bass lines aren't terribly challenging, so it's easy to sing over them. As I started playing Sabbath, Stone Temple Pilots and Iron Maiden, it became more of a challenge to mesh the two roles. (However, if you want a real exercise in frustration, try singing and playing Van Halen's "Mean Street". Yeesh...)
Ultimately, I think you just have to decide what will serve the song best and do that confidently. Most people won't cluck their tongues if you don't nail two parts simultaneously. Have fun!