It was, in fact, flapping like mad! The shop didn't have exact replacements so I've decided to take a gamble and stuck in some $0.49 school ruler from office works... It fit like it was meant for it! I just had to shorten it a bit but the width was literally made for the pocket.
I've had one session with 22+ knots and no flapping at all, the kite flies smoothly as. I just wonder how long the ruler will last but with the price of $0.49 I can't complain.

I've even done some testing on it and I can bend it both directions to full "U" shape so in my opinion, it will last few months at least.