Falconry transmitters are used by a few UK rocketry enthusiasts.
They are very small, and have a decent range, but the quality does come at a price.
I very much doubt that you will be able to design a stable transmitter, that has a long life using a tiny battery pack, and is small enough not to annoy the bird.
Don't forget that birds need feathers for flight and guidance, and a heavy transmitter could easily yank the tail feathers out when the bird is doing high G avian aerobatics!
Do the bird a favour, and buy the correct kit.