Imo, a psychiatrist is key for someone like you who has had difficulty finding a SSRI that is effective sans side effects. Neurologists and PCPs rarely have the experience to do much more than the bare minimum when it comes to psych meds in most cases again, imo.

That said I have found many medications are changed prior to reaching the maximum dose and if there is even a partial response I would prefer to finish with one before moving to another. You could also consider augmenting with a second medication if one just isn't enough.

Good luck.