Hi fellow glasses and contacts sufferer lol....
I agree that your glasses case is a bit heavy. But do NOT put them in a soft case, baggie, etc - it's not worth potentially busting your glasses on trail.
I am going to save your day right now. I discovered this glasses case thanks to Mari Johnson's PCT vlogs - she just randomly bought it off Amazon and used it, and I was curious so I got one too. It's a beast of hardshell plastic case that weighs a mere 1.9oz (and that's with a wiping cloth inside). I would never backpack without it, since I got mine. You're welcome.