If I may - a few questions then some speculations: When you say "Swan on the right side" were you feeling the pain in the back (left) leg or the front (right leg)? Why do you think that the Swan created the pain - surely you were doing both sides of the Swan and you were doing other poses too? What were these other postures? Are you only correlating the pain with the emotional release from the Swan on the right side: if so - why do you think that these are related?
You have seen specialists who have the advantage of knowing you and working with your body. While you have given us a very detailed description of your sensations, the ability to work out what is causing your fire is definitely limited by being restricted to internet interactions. But, let's make a few speculations and suggestions, with the proviso that all this should be taken only as such - guesses: check it all out for yourself.
First - the cause. I would have to agree with your other health coaches - it sounds like a nerve has been stimulated: such burning usually is a signal of an unhappy nerve. Pressure on the nerve could come from a variety of sources: tight muscles impinging upon the nerve, disc problems, inflammation, etc. You may have work to through several layers of potential causes before finding a treatment that addresses the right issue.
Second - what to do? If the problem is from a tight psoas, then massage might help, but I have heard conflicting advice regarding this. Some therapists feel that it is just not possible to really or effectively get at the psoas: it is too deep. Others believe that they can get to it. Certainly it can be stretched, but if the nerve is already inflamed, stretching or even massaging the psoas could make matters worse. You may have to leave it alone for a few weeks so that the nerve quiets down, and then work poses like Saddle or Dragon to stretch out the psoas. Since the Swan does work the hip flexors of the back leg, perhaps this is why you had a trigger for the pain after doing Swan. (This is the reason I asked about which side you are experiencing the pain in: left or right. If it was on your right side, then Swan on the right side was not stressing the psoas, and thus maybe the psoas is not the problem.)
Nerve compression due to inflammation? Unlike your MD’s advice, I would not recommend ice. This used to be the standard approach to treating inflammation - the acronym is RICE: Rest, Ice, Compression and Elevation. More recent studies of cryotherapy
has shown that ice is actually not effective. Today the process recommended is called MCE: mobilization, compression and elevation. If you are talking about your psoas, it is difficult to apply compression and elevation here, but gentle mobilization may still be effective. There is another option to reduce inflammation: earthing. You can read more about it in his newsletter article
Disc problem? Unfortunately, here too there are so many possible causes. You can read this article on sciatica
and try some of the suggestions there, like spinal flossing, to see if they help.
Finally - until you find the cause, a cure is hard to prescribe. However, even without knowing the exact cause there are some general things you can try: stay active, but don’t stress the area where you feel the “danger” lies. If you feel Swan was the cause, don’t do Swan! But mobilization is important, so gentle yoga may still be good for you: Cat/Cow flows, Spinal Flossing, etc. Earthing is a great thing regardless: even healthy people should spend time every day getting grounded. Diet can also help to reduce inflammation (think turmeric!)
Let us know how it goes.