eleonorang I agree with everything @rjgilsdorf said in the comment above. It seems like you had the contract so they can't just change it.
Personally I would politely point out that you've accepted the original offer with original price which was fair taking into account the deadline and complexity.
If they don't want it, then it's fine, something better will always come along
Even if it was an honest mistake of the said PM, you shouldn't be punished for it. There's a contract: they said they gonna pay X dollars, you agreed. Simple as that.
Of course, if you really like the client you might negotiate, but I wouldn't do it for half the original price. I'd give them 10-15% discount at best.
I would also avoid clients with poor organization skills in the future. Nothing good comes from it in my humble opinion.