How to isolate H$_\alpha$ from Nitrogen doublet emission lines
I would've posted this as a comment, but I lack sufficient reputation.
I like your idea of fitting three Gaussians, but I think you should try fitting with one large positive Gaussian and two small negative Gaussians.
My physical justification for this is that I would expect a lot of self-absorption from the H-alpha line, which should be well-modeled using the larger negative Gaussian. I'm not sure if the other notch is absorption from another line, but it also looks like a negative Gaussian would fit well there.