How can I identify and source an old faucet cartridge
That looks like an American Standard brand of faucet. Probably same stem as a Hampton (pic below) or Williamsburg style faucet. The part number for that cartidge might be 994053-0070A pictured below. The cheaper version would be Danco 10472 4Z-24H/C Stem for American Standard Faucet. You should try to call American Standard (800 442 1902), and you should try to send them your pictures (above). And read the fine print on the warranty from American Standard.