The request for CoO for documentation is BS la. All they need to keep is a proper invoice from you.
What you should do is to send them a bill/invoice, and estimate the cost of using your work and include the target audience size etc.. remember it is used a TV show, so pretty much the audience hit is very great. After that send demand for payment. No need to play their stupid run around game. Bill Mediacorp direct. They will push to production house, but you can ignore and just say, you saw the use by mediacorp. So your bill goes direct to the end user. They will have to claim their damages from the production house themselves. Do your normal job of sending payment reminder, final reminder, then lawyer demand (with late fees, and admin fees, lawyer letter fees). Then intent to sue. They will move, trust me.
All correspondence in black and white.
And I sure hope you have registered a business in your own name. If you do not have a registered business entity, it will be very hard to get all these done.