You can force the disclosure through court by a hearing in probate court. BUT, that is unrealistic for the cost of such an action. Things become more serious if he is sued and it includes attorney fees.
Your attorney should be able to tell you about other options which include, distributing to heirs who have provided the SS numbers and holding is in escrow.
However the attorney should be able to file without it, by including a note that the person refuses to provide the SSN. The attorney should provide a W-9, along with a note indicating that under IRS regulations, failure to provide a SSN can subject him to a fine from the IRS. (it is nominal, 75 dollars.).