It might be good contrast to the Kelvinverse movies and general pop culture image, if they actualy start out professionaly distant. Like they clearly have two distinctly different viewpoints and openly express their disagreement but are too professional to be openly become hostile or shout.
Maybe even end their first encounter with both expressing relief at no longer having to work together and being doubtfull about having to do so again. So the fans can have a big "oh if only they know" moment and look forward to their next encounter.