I would respond to AAs letter / email simply stating that you require reinbursement of all the costs you lost because of the mistakes made by their staff. And remind them that they have already admitted to you that it was their mistale and you need to be made whole.

Compensation is different from reimbursement. Compensation is something over and above your loses. If you lost $2000 and they pay you $500 that's not compensation because you are still out $1500!

Quote them the timatic requirements (as quoted above) and also anything you have in writing from the Aruba authorities / screen shots of their websites to reiterate AA were the ones to make the mistake in denying you boarding.

I wouldn't mention lawyers or courts at this stage..

If they then reject you again then escalate it to the executive level. When you do that then mention you will take it further externally but again don't specifically mention court. You want resolution not a battle.

Only if they still refuse to make you whole would I go down the court route.