Thank you for keeping us informed and posting the videos.
1. For my capacitor tuning (L-network, not a loop) I use an RC servo. The cheap SG90 servos (less than $2 on eBay) will do 180 degrees rotation in fact and the angular accuracy is good enough. Less noise but still not silent either. But you can drive the capacitor shaft directly since there is no need for gears and therefore it is easy to make a tiny sound proofing box for them. RFI into the servo was fixed with a simple toroid for the control cables just at the servo entry. If wanted you could cut power to the servo with a mosfet to save energy (a few milliamps) when not moving.
2. For tuning, you can use CW and reduce power by using Ted's method (see https://groups.io/g/BITX20/message/67662).
You would need to change the cat_lib.ino library a little. For example you could create an AM mode in the CatSetMode method that sets an AM flag to use later in the CatSetPTT method.
In the CatSetPTT method on a PTT ON command, if AM mode flag is set, you could use a startTx(TX_CW, 0) followed by cwKeydown();
On a CAT PTT OFF command just reverse with cwKeyUp(); followed by stopTx();
That's the logic I use for my tuner.
That way you don't need to change the loop tuner's code.
Hope that helps,
73, John (VK2ETA)