harpwood Hey there,
first of all: congrats on finishing your game! Getting the last pieces done and pushing it out the door is always an exciting (and often stressful!) part of the process. 😀
Of course you can use the Starling and Gamua logos – thanks a lot for crediting Starling your game! In fact, there are a few nice badges you could use, to be found here. As for the Starling bird itself, here is an SVG file with a few variations of it.
I hope that helps! All the best for the final touches, and don't forget to let us know when the game is finished!