In the real-world, it is called professionalism. Hopefully both director and designer will have it. I have disliked (and even hated) colleagues in the past. Does not stop me from working with them professionally,
Whilst the SM is there to lubricate the lines of communication, the role is not to manage the director or the designer and mediate their relationship. You can help to reduce contact between the two - and even sanitise nasty comments that come through you "Director says XYZ looks like sh!t" becomes "Director would like to re-evaluate XYZ as they are not cognisant with their artistic vision" etc.
Generally there are pretty clear reporting lines within a company. IF your production is suffering significantly owing to the tensions between these two parties, then you need to talk to their higher ups - the artistic director, company manager, whoever. Not to "tell on them", but to report your concerns. This person will then make the call as to what they want to do with it. If you inject yourself in the middle of it, you run the risk of ruining your relationship with one or both parties. That is not good for anyone.
At the end of the day, if the production is at risk of failing due to this pair (or there is risk of people getting injured, hurt or ...), and you have aired your concerns and nothing has been done, then you have 2 choices. Put up with it, or walk away.