Is it permissible for a Mubahalah to take place between a mosque's imam and a board member concerning an administrative dispute, and would the imam's invocation of Mubahalah against the member result in harm befalling the member and his family?

You did well by advising him and clarifying the necessity of brotherhood. Continue to invite him with wisdom and good counsel, and explain to him that escalating a minor disagreement is not befitting a Muslim, and that resolving disputes is done with calmness and tolerance. If he returns to what is right, then praise be to Allah. Otherwise, his prayers or curses will not harm you.