1. Presuming that OP accurately quotes the two UA replies, he was not denied because UA doesn't reimburse for the visas, he was denied because he failed to document a receipted expense. There's nothing else to argue about, so UA closed the case. No receipt = no refund.

2. But, OP does have a receipt. Turkey chooses to create the visa & receipt in one item. That may not be typical and it may be annoying, but it's what Turkey chooses to do.

3. When OP was initially denied, all he needed to do was submit a photocopy of the two visas which presumably show the date, location and amount, and he would by now have his $40. But, he didn't. Why UA would call him to "discuss" something which seems quite clear, is no more relevant than whether the commitment to pay the $40 was made by a UA employee or contractor.

4. If this is about the principle of the thing, OP should submit a new claim, not mention the old one, properly document the claim and hope that nobody refers back to the old one. If this is about the money, then drop it, unless OP makes less than minimum wage, this thing has cost him a lot more than $40.