He would, but he would never ask about it. They both seem to keep it pretty professional with the understanding that deep down, they're friends. I imagine they catch up a bit during that yearly coffee (which will always be canon to me), but if Jim started cracking or looked visibly distraught, Bruce would talk to him. They're friends, and when Batman isn't too busy being used as an all-business-the-crusade-never-sleeps allegory for men as gods, he can stop the mission for a brief moment to help Jim out.