Doppler shift?? That must mean your band and the audience is continuously expanding, like the universe! (assuming your sound guy thinks it's a red shift and not a blue shift...if he thinks it's a blue shift, you and the audience would implode and end up as a white dwarf). You should probably ask him which color the shift is. And while you're at it, ask him where he gets his drugs because he obviously gets the good stuff!
Playing through your rig is fine. I have been doing that for 40 years of playing bass. It leaves the PA more pristine for the other instruments and vocals.
Now, if you had a monster PA with lots of wattage, and a bi or tri amped PA speaker system with subs, that would be another story, but based on the equipment your band has, you are doing the right thing by just playing through your rig.