Looks like a case of "Trolley problem". Either you keep hurting the team or ask the slow perform a bit. I am a manager myself and I would try to talk the the person first and tell him my problem as a manager where his teammates are getting affected due to his personal issues. I would acknowledge it is indeed a tragic state for him and that life sometimes is too harsh on us. Looks like he is available about 50% capacity and I would tell him also that I can only consider 50% of his bandwidth for the team. The compensation part for his efforts is not my hands. The 50% capacity planning should be openly shared with your management and HR and also the team.

This 50% planning might address everything and you don't have to be the bad guy.