For what it's worth, I have a teacher that's very similar to yours.
I have only been playing for 4.5 months, and my teacher recommended that I strictly alternate when playing ANYTHING on the bass. It sucked at first, but now it's becoming almost second nature.
My teach even goes as far as making me play lines starting on the index, then doing the same line starting with the middle finger, essentially playing the inverse alternation. It gets really tricky with more complicated lines, but it's nothing I haven't been able to overcome with repetition.
I can almost see the payoff by using this method, so I say stick with it, you'll be glad you did.