Broken capillaries at the base of my lower eyelashes - can these be treated? (Photo)

Not to worry. The use of low intensity electrodessication has a proven safety and efficacy record gong back decades. The treatment area is numbed, and then a low current is delivered through the surface of the skin to these tiny vessels through a superfine needle that closes them off permanently. The eyeball is not touched at all--simply the surface of the skin above the unwanted vessels. I have personally been using this well-known technique for several decades with very gratifying results.
A note of caution: Excercise a heavy amount of buyer beware when being hawked expensive, "bells and whistles" light or energy-based gizmos, since the hard science to back up their use lags well behind the heavy marketing hype and social media buzz to promote them. Hope this helps and best of luck.

That is no problem getting the broken capillaries zapped near the eyeball. It is commonly done and easy to undergo. It may need 1-2 treatments though.