Most likely, none. Your contract is with the travel agent, and it's standard practice in the industry that they are your point of contact for any changes you wish to make. When the agency says that you can contact the airline, it doesn't really mean much other than that the agency won't stop you from contacting the airline. It doesn't guarantee that the airline is willing to handle your request directly.
From an airline's point of view, reducing staffing requirements by shifting these requests elsewhere is one of the attractions of dealing with travel agencies. Some airlines may still allow you to contact them directly for cancellations or changes, but many do not or will have limits on the kind of changes possible.
If you do want to try again with the airline, you may have more luck contacting them by phone rather than using the website. The staff may have a little more leeway than public-facing websites. But even then your chances will vary with airline policy, and perhaps with any frequent flier status you have.
Ultimately, accepting the possibility of cancellation/change fees imposed by the travel agent is a trade-off you make when booking via a travel agent.