why would you want to shock people with how different the retro Doctor is to begin with? you should stop thinking of it as less of a Classic/New distinction and more of a different Doctors distinction. like, if you're playing Six then of course there's going to be some tension between Six & Rose because her Doctors are very different from him....but there also would be tension between Six & Tegan or Six & Victoria because again, very different Doctors. and with some New Who companions, there's less of that culture shock to begin with: like, I'd imagine a season 9 Clara would be super chill with a Classic Doctor (because s9 Capaldi has more Classic Doctor traits) and conversely, Grace Holloway would be super chill with a New Doctor

just stop thinking of series differences and play the Doctor like he's the Doctor. Points of identification will inevitably pop up because the Daleks, the Time Lords, and the Cybermen have been around for fucking ever. If you want to do any culture shock sort of thing, focusing on one of those crossover aspects might be your best bet. like, for example, how would this Doctor react to the idea of the Time War or the thought of the current state of UNIT?