You could go with a hard money lender.
I'm assuming the closing on 10/13 is a sale by you and the 10/24 closing is a purchase by you?
Your contingency is a financing contingency, IMO. Now, the seller (quite reasonably) wants the contingencies sorted out as soon as possible, but I'm not sure that waiting 4 weeks for that contingency to lapse is such a hardship on the seller.