I'd advise against it. It would probably be strong enough. But, with rope this thin, the kink where the verticals run across the loops will become very sharp. As you tune, that will put a lot of extra wear on the rope. You risk damaging the sheath on the verticals because the small contact patch will make for a huge amount of friction on a very small surface area.
Think about it like this: if you could make the loops on the crown ring out of something very thin, such as steel wire, the loops would end up cutting right through the verticals. With 3 mm loop, you are not that far away from that happening, I suspect.
I've never gone below 4 mm for the crown ring. These days, I use 4.5 mm for djembe, no matter what size.