Your lab is correct. You can most certainly pull (or push) E-6 films, and quite a bit. It is one of the main reasons that I choose to use them over negative film. They are a color medium with which you have much of the control you have when using black and white film. Usually, they maintain perfect color within a one stop range from 1/2 stop pull to 1/2 stop push, with some exceptions holding perfect color at more extreme pushes and pulls better than others, and a few doing it worse. Outside of that range, you get minor, correctable color changes, with color usually being acceptable/correctable down to about a 2-to-2-1/2 stop pull.
What you do with exposure is entirely independent of what you do with the process. You don't need to push or pull just because you exposed a certain way. Just like with black and white, your exposure determines largely how your low tones are rendered, and your development determines largely how the high and mid tones are rendered.
In any case, I would not re rate the film. I would simply manually over or under expose it as needed.
Using sheet film, or dedicating entire rolls to the same subject, are helpful with these methods.