Hello, I'm Dr. Deb and will do my best to help you today.
I'm sorry that your Lhasa may have been injured by this other dog.
Systane eye drops are appropriate to use in his eyes as long as they don't contain vasoconstrictors, such as naphazoline; I'd advise that you double check the label and it if does contain this ingredient, I purchase a product without it. These products are usually labeled as "taking the red out" or for "red eye".
If he has an obvious wound or injury, then Clavamox is an excellent drug for this particular issue. The dose would be 62.5 mg for a 10 lb dog, 125 mg for a 20 lb dog or 250 mg for a 40 lb dog, given twice a day with food (if at all possible) to avoid stomach upset.
Any skin wound can be cleaned with dilute over the counter Betadine or Iodine (to the color of weak tea). I'd avoid getting any into the eyes, though.
Another product which you might consider would be Vetericyn Spray which comes in a product for wounds and one which is an opthalmic gel which can be used for the eyes.
It might also be helpful to take his temperature; normal is between 100 and 102.5. If it's elevated, then this may be the reason why he's not feeling well.
If he continues to remain "off" or if he's keeping the eye closed or squinting with it and/or there's a significant green or yellow discharge, then there may be a corneal injury which would require drugs best dispensed by your vet. A vet visit would be prudent under those circumstances.
I hope this helps. Deb