I have bought a Turkish Airlines plane ticket through online travel agency https://www.smartfares.com. The ticket included checked baggage ($20) but at the check-in I discovered that the airline "does not see" any baggage in the reservation. Therefore I had to pay more than $130 for my baggage at the counter (which treated as "overweight").
The agency support line confirmed that I paid for the baggage but claimed it was the airline fault to not include the baggage option so they will just refund "unused" $20:
We regret TK failed to action the add on for your fights to this Istanbul airport. We hope you are safe where you are now. Thank you for letting us know, we have reported the issue. We have refunded you for the failed add on.
After that they did actually refund $20 but it is definitely not what I want (as I paid excess $130 for "overweight").
Is there any good chance I can get a refund of remaining $110 from the agency? From my understanding, it should not make any difference for me if agency or airline (TK) "failed to include the baggage". As I paid money to the agency I expect them to deliver me and my baggage to the final destination. I should not be financially responsible for any communication problems between the agency and their contractors (in this case Turkish Airlines).
In your opinion, what should be the communication with agency (or airline) from my side to maximize the chance of getting the money back?