OTOH, if you put it in a box but the cw beacon still whines even after you allowed the downconverter to settle, there might be a problem other than air draft. Can you make a screen video of your sdr software showing the cw beacon so we can see and hear what's going on? Please also list all equipment involved, like dish-size, lnb, receiver.
As a reference how it should sound (well, I think it should sound like this) , see https://www.<mycall>.de/uncategorized/qo-100-beacon-snr/
BTW, place the box where vibrations are minimal, preferably on a piece of foam. I once operated it on my operator desk and even bumps from your fist on the desk can be heard as small frequency jumps when you listen to the cw beacon.
Remember, a frequency variation of 10 milliHertz results in a 3.9Hz shift after multiplied by 390. You will hear that. The crystal in the tcxo after all is a mechanical device. It wants to sit still (and flat) for optimal work.