How can one absolve themselves of liability for an advance payment made for a machine that was damaged and stolen due to war, given that both parties have suffered harm and the customer is in need of the money?

What transpired between you and the client is a manufacturing contract (Istisna'a), and the goods remain under the manufacturer's guarantee until the client takes possession of them or is enabled to do so. Since the client was unable to take possession of his machinery due to the factory's destruction, the goods are considered damaged while under your guarantee, and you are obliged to refund the paid amount ($10,0) to the client.

Evidences:

Muhammad ibn al-Hasan (definition) al-Shaybani: "If the buyer does not take possession of what he bought, then whatever perishes from it, be it little or much, is from the seller's wealth, because it perished under the seller's guarantee before he delivered it to the buyer." Al-Mawardi: "If the sold goods are destroyed while in the seller's possession before the buyer takes possession of them, the sale becomes void, and they are under the seller's guarantee, and the buyer is entitled to reclaim the price." Shari'ah Standards (Islamic Accounting, Auditing & Governance Standards): "The manufacturer's liability is discharged upon delivering the manufactured item to the client, or enabling him to take possession of it... If the client refuses to take possession of the manufactured item without a valid reason, after being enabled to take possession, it becomes a trust in the manufacturer's hand."

Based on this, as long as the manufacturer has not completed the manufacturing process and the client has not been able to receive the item, the manufacturer's liability is not discharged.