Having the exact same trouble with an S4 with the giant Zerolemon battery.
Lokki, I don't think you've seen one of these - the NFC antenna on the back is absolutely HUGE! It's about 3" square (7 cm square, for the metric-minded) and swiping vertically, horizontally, with the "rolling" swipe - doesn't matter. This ring does NOT respond. No, not even with a completely "away" swipe like you're describing.
I'm thinking of just putting a cheap NFC sticker on my belt or something, and swiping THAT instead. The standard 203 chips and even Micom chips work every time, even with the case on. Not as elegant as The Ring, but at least it WORKS.
Unless there's some magic fix for this, I'll have to call it a day and return these.