What kind of claim is the receiver considering? If there was no damage to the freight, then your cargo insurance policy doesn't apply. Unless you have a policy that covers missed appointments (which I am pretty sure does not exsist) the only policy that will cover this is your own "good will", in other words - out of your pocket. You should bring to their attention, the clause in the Bill of Lading that talks about "in time for any particular market.....otherwise than with due dispatch". Is there any notation on the Bill of Lading that indicated delivery must be made on a certain date? If so, the trucking company must have been aware of it when their agent, the driver, signed the Bill. If there wasn't such a notation, then it isn't a part of the contract of carriage. I have had this same situation occur. The carrier accepts the freight, agrees to the rate, agrees to make the delivery on a certain date, you confirm all this with your customer and then WHAM, they deliver late for reasons sometimes reasonable and other times not. I know what replies will flow from your post, either "if it was so important, why didn't you ship it as a truckload"? or "hey, it's LTL, what did you expect? I am sure every broker has heard both of these lines more times than they can count.
From my experience, the best defense against this is to use only your most reliable carriers when a shipment requires a timed delivery, and to follow-up closely with them while the shipment is in transit. Even then, screw-ups can occur, but at least you have done your job to minimize them. The long and the short of it is - how important is this customer to you and how much are you prepared to eat of their so-called "claim"? You could ask the carrier for some assistance, but if they have already lied to you - well we all know what their answer will be.