My biggest concern about swinging on any fitness rings is the buckle slipping and shredding the strap which will jar one of my shoulders into oh, so fun seperation or shoulder tear.
If your swing is mild like a compulsory swing (little kid swing) it will probably hold. If you look at regulation ring straps, it is just a buckle that sleeves through rather than grinds teeth in to.
One of the biggest concern about swinging on fitness rings would be on what they are mounted to. If they are mounted from the bottom of a support beam, they could easily crack it or tear right out. From the side might work but if the strap overlays across a beam it will probably fray at some point.
I'm pretty sure some mild swings like uprises or kips/butterfly would be doable. Not gonna be smart to really start swinging on them. A mild swing of 3g is probably too much. I have hung and swung on them just to loosen my back a bit.
Attend the gymnastics class. The feel of sprung rings is far different than hanging rings. And of course there are mats to land on if you peel and land on your anterior.