Any treatment to make bulging eyes look more sunken & eyelids less visible? (Photos)

For significant improvement, hooded eyelids require surgery if there is enough skin. For nonsurgical treatments: Radiofrequency (Thermage or Venus Legacy/Viva), injectable fillers, and a Botox brow lift can help contour the eyebrows and lift the eyelid skin. I recommend getting a formal evaluation with an experienced cosmetic dermatologist. Best, Dr. Emer.

I'd first recommend that you have a work-up to rule out an endocrine condition - in particular, thyroid issues.
My suspicion is that this is hereditary - meaning "you've got your dad/mom's eyes"
Assuming no endocrine issues as I noted above, I would consult with a make-up artist for recommendations, as I agree with you that non-surgical options are best.

If this is a new finding, this could be a sign of changes in thyroid hormone function. Assessment by an ophthalmologist and endocrinologist would be my first recommendation. Some patients have naturally "bulging" or what we would call proptotic eyes due to the shape of their eye socket. However, changes in the volume of the eye socket due to excessive growth of eye socket structures can be caused by changes in thyroid function.

You have prominent eye, a feature that often goes along with hyperthroidism. There is a surgical procedure to remove some of the fat from around your eyeballs so they are not so prominent, and now I see on TV that there is a medication that claims to do the same thing. I suggest you seek out an occuloplastic surgeon who has experience with this entity.