I had lots of problems with conjunctivitis. The GP nurse told me to get Baby Shampoo. This does not sting eyes.
Well at least not much.
You use just a few drops ina wash-basin of warm water and wash the eyes as much as possible.
This worked om me.
Any kind of eye problem on the dog, grit, flies, weeping etc and I do this. So far has worked out fine.
All that is happening is that the liquid allows the water to "wet" the eye more easily.
I use a small sponge and just go for it. He really does not mind.
And he would moan like mad if it hurt.