I would imagine this would be something you would have need to have had exempted at time of ticket issuance - if you were eligible at that time - not at time of check-in with LA, especially on a ticket issued by another airline. Once a tax is collected, and in this case shows as used, it's paid to the government or authority who the airline is collecting it on behalf of.
Is it the normal practice for to claim a refund of this tax at check-in? (assuming LA issued ticket on an LA flight).