It stinging is a good indicator straight up mate. Cetaphil is designed for dry skin, not an open wound. Have a look at the ingredients, it's alcohol based.
This shit here is fkn good shit for fresh tattoos. It cleanses and has a mild local anaesthetic.
In saying that though, personally I pretty much dry heal.
Post tattoo, old mate hot water towel pats the tattoo down. Not boiling water, but pretty warm. An hour later I'm home, pumping hot shower while drinking ice cold cider. (The cider being the most important part).
I'll use a soft body wash to hand wash the tattoo down, one with an Alo Vera or similar base.
The only time I'll use the paw paw is if an area is red/ inflamed & only just a light film. No need to over gunk.
Cetaphil is a great moisturiser to help nourish when dry, but again, I'll hardly ever use it and it doesn't necessarily have to be Cetaphil. I've only been using it recently on my hand & knuckles as it was healing hard, and the Mrs had it handy. Don't stress about the ink either. Sounds like ya just copped a belting and it's weeping a little.
I ain't no expert on healing, and we all have different ways. This way just works for me.
Hope this helps.