As you well know from your relationship with your yoga teacher, it is best to ask this of her/him. This person can see you and make the requisite corrections to your asana practice, teach you how to work with light and breath, and provide you with the necessary changes in nutrition and lifestyle to manifest your well being.
Vira III and Ardha Chandrasana are poses where weight-bearing is done on only one limb while the torso is 90? to the support. Therefore it is completely expected to have additional physiological challenge with these poses when contrasted to Vira I/II. So this is quite common and may not have anything at all to do with flat feet.
Also common, especially in athletes (cyclist) is tension in the IT Band. Keep in mind that the IT Band is a stabilizer and therefore it is best not to over stretch it.
An IT Band issue would more directly correlate to the feet if you were a runner. However it is also possible your cycling mechanics are not helpful. Have you considered a Fit Kit?
Eka Pada Rajakapotasana may work three of the eight hip actions but may also place strain on the knee and sacroliac joints. So that may not be the best "remedy’ when facing a therapeutic issue.
Cycling obviously works flexion and extension. A mindful asana practice for a cyclist would place emphasis on other hip movements (toward balance) like adduction, abduction, internal rotation, external rotation, and extension of the femur away from the hip joint. I have no idea what your particular Yoga practice involves therefore “4 to 5 times per week” does not help me.
In Virasana the medial condyles of the knees must be equidistant from the floor or in a plane parallel to it. I mention this because Virasana is one of two poses (the other being Padmasana) where students can “come into the pose” but do so without integrity and place the knees at risk both in the moment and over time. Since I cannot see your pose I cannot tell you if that is the case and therefore cannot determine the integrity of the pose for the knee joints. So your discomfort in coming out of the pose (which I presume is in the knee) may be alignment oriented.
Flat feet are addressed through actions in standing poses (again a task for your teacher) and through manual stimulation of the soles in virasana.