Well , you are right in every aspect of your justification and yes you and your husband do deserve an apology from your brother in law , as right now he is the one who is behaving unlike a mature person and after so many attempts by you and your husband for patch up, any understanding and mature person would have called off the argument and also would have tried to pacify his son instead. But , from what you have mentioned , i do not see it coming any time sooner. And right now you should not feel guilty and doubt your credentials as a good human being , because such things happen in life and i guess you have done more than enough to see off the whole tiff, what can one do if the other party is behaving so childlike and going to the extent of banning someone from family gatherings. I feel this is truly uncalled for, and may be your mother who is the head of the family should intervene and convince your brother in law and his family to let go off the whole issue.
As far as your brother in law's demand from you to seek professional help and produce some kind of mental fitness certificate before you shall be allowed to attend family gatherings is concerned , this in itself is a laughable childish ploy by him to kind of cause emotional hurt to you and to mentally disintegrate you ,so that you become upset and he can avenge his son's insult in this manner. Such kind of behaviour , just shows how disillusioned and unreasonable your brother in law is acting, i would suggest that you do not need to heed his comments at all , in fact they should not be taken seriously at all.
So, i suggest you to talk to your mother and your brother & sister about the whole thing , may be they can intervene and seize the whole issue, but as of now you and your husband have tried enough to appease your brother in law , but to no avail, so you shall not try to apologize again, as right now you people are the ones who deserve an apology, after all you need to guard your self respect also. Also as time passes by, it may bring sense to your brother in law's perspective and he may himself like to close the whole issue.
I hope my answer serves your query according to your satisfaction.