@georgejoseph, I'm back and ready to share what I know about Baker's cysts, especially how to get rid of them. I agree with your surgeon....don't introduce anything foreign until everything has healed. My surgeon cleared the cyst in the armpit of the knee as you call it so creatively, during the surgery. It never returned or refilled.

I am now getting ready for a TKR in August on the other knee. i have had the quite annoying Baker's cyst drained twice. When you look at the images taken during the aspiration you can see how the fluid-filled cyst can push other tissues out of the way.....thus causing pain.
Cysts like this usually fill with fluid that our body creates to try to soothe or overcome a damaged area.

If you cannot have it drained, you can remove it by pushing it up and out of the knee area. Once you have moved the fluid up and out into your upper leg, you will just pee it out. You might be helped by a Myofascial Release (MFR) therapist. I see mine twice weekly and without her, I would be in a constant battle with "Mr. Baker". For more information on MFR please follow this link.
https://connect.mayoclinic.org/discussion/myofascial-release-therapy-mfr-for-treating-compression-and-pain/
And I have one more helpful suggestion. Practice the Yoga pose named "Leg's up the wall". I do that at least twice a day. There will be instructions on youtube. It is a great way to keep open all the valves in your legs and let the fluid out.

Before we go any further would you please tell me more about the fluid (swelling) in the armpit of your knee. How does it feel? Hard, soft like a pillow. Where is the pain centered? What other options have you explored?

May you be free of suffering and the causes of suffering.
Chris