I think you should let the first airline off the hook since the damaged suitcase is already replaced by the second airline. Not sure if it would have costed more to repair the damaged suitcase twice or replacing it, but had the first airline repaired the damage it's responsible for, the second airline may not had to replace your suitcase for it's responsible damage. In essence, the second airline compensated you for both it's responsible damage as well as first airline's responsible damage by replacing your suitcase. Therefore, I don't think the first airline really owe you anything other than the inconvenience it created by delaying the arrival of your luggages.

If you have nothing better do with your time, you can refer the first airline to the second airline to see if the second airline wants to be reimbursed for the compensation it gave you for the first airline's responsible damage. I doubt the second airline would have the time for this, though.

LAX