From my other comment: Never owned the wireless version, but on the Pro + GameDAC version, the serial number can also be found on the bottom of the DAC. There may be a serial number on the base station as well.

I read through the post again, and I honestly have no idea how OP managed to wear the sticker off, unless it was through poor usage. There is a ski-goggle style band that wraps around the metal headband (which the S/N sticker is on), but does not touch the sticker in any way (first picture, different model but similar design). It doesn't matter how you wear the headphones, it just won't make contact. I've gone through 4 pairs over the years (even bought a second hand at one point), and the longest one I've used was about a year. The stickers were clean and visible in every pair. OP said they recently bought the headphones (assuming a few months ago from the box comment) and the sticker already wore off. As I already mentioned, the stickers never wore off even after a year of use, and the stickers do not make any contact with anything even when you wear them.

It's standard/required to provide a serial number along with a proof of purchase whenever you go through any RMA. I've gone through RMA from NZXT, Razer and Steelseries multiple times with no problems, and they all required a S/N for a valid replacement.

I know Steelseries can be shitty for some of their bad business practice like completely neglecting the design flaws of the Arctis Pro, but I never had problems with their RMA process. I had my Arctis Pro replaced twice without issue, no questions asked when it came to the design flaw. I know it sucks, but this is on you.

Edit: Never owned the wireless version, but on the Pro + GameDAC version, the serial number can also be found on the bottom of the DAC. There may be a serial number on the base station as well.