I had the same issue after I recently bought the AirPods Pro 2 on Apple website. I tried everything: unpair, forget, repair and nothing worked - the engraving didn’t show on my phone. I engraved the case with my email address. It delivered the next day.
As I was still within the time frame to return, I returned the device and then bought another one on the Apple Store app. I added a Memoji engraving. This is only possible when buying on the Apple Store app.
First, I noticed that with the Memoji engraving, delivery will take around 2 weeks (although I received mine within one week). I also noticed that the AirPods shipped from China.
I received the AirPods today and as soon as I paired, I saw the engraving appear on the screen.
I have checked model number and it’s A2968. Same as the one I had to return.
Note that I live in the UK. Given that the first one took only 1 day to deliver, I think it was engraved in the UK.
The second pair was shipped China and I guess this made the difference for me.
I know there be others who may have had a different experience.
Well, I’m now a satisfied customer. The only reason I decided to buy these was because of this added feature.