Man, they really fcuked you up and telling you you have to sort it out by Monday is an even bigger joke.
The fact that you self declared in writing is good though. Have they given as to why the background check wasn't satisfactory to the company, or was it because of your self declared disciplinary action?
I wish you could name and shame the company, but probably not a good idea, as this would prejudice any action you may have against them.
I think in this case the best thing you can do is try and negotiate a settlement with them that covers the cost of your family relocating.
I would definitely seek the advice of an employment lawyer though. Tanner De Witt are a solid outfit. However if you do want the name of another law firm, can you PM me.