What Can I Do To Correct Big, Bulging Eyes?

I suspect that getting tested for antibodies will be fruitless here.  Obviously, follow the recommendation of your doctors.  It sounds like this has been a very long term problem for which you have be relatively well worked up for.  You look like you have maxillary hypoplasia.  I would be very careful about having craniofacial surgery for this.  Orbital rim fillers are now very popular for this.

The most common cause of bulging eyes is thyroid eye disease. I assume that you have had thyroid function tests and thyroid autoantibody tests (a small percentage of patient's with the disease will have normal thyroid function tests and elevated antibodies). In addition to the bulging in the left eye, there appears to be some eyelid crease asymmetry, which may be compounding your concern with asymmetry.  This is more evident in your second photo. I would suggest a comprehensive evaluation with an oculoplastic specialist.