If the employer insists on treating this as FMLA (which they need not; given the fact pattern we've been given - they could easily give the OP all the time, paid or unpaid, they want to without activating FMLA at all) I think the best solution is to try the doctor, if any, who is treating the younger siblings. Although there is, as I said, a precedent for an employee to use FMLA for grief-related depression, I am not recommending that the OP attempt that unless his doctor believes he needs it.
But the truth is, while FMLA is not inappropriate under the circumstances, it is also not the only way that leave for the OP can be accomplished if he cannot get a doctor to agree that FMLA is warranted. FMLA is a very specific form of leave with specific parameters and which can only legally be used in certain circumstances. Other forms of leave exist that have no government regulations restricting them and the OP's HR really needs to be thinking out of the box. There is absolutely nothing wrong with or illegal about an employer saying, "Joe, we know you've got a lot of issue going on with your family - take x weeks/months and come back when everything is resolved and your job will be waiting for you - we've got your back". Up to them whether that time is paid or not paid. But it doesn't have to be FMLA, so maybe what the OP needs to be doing is talking to them about alternate possibilities.