AirPods connect automatically on iPad, but having to reset from base on iPhone

I have a bit of a unique issue. My AirPods Pro 1st generation had a problem so I decided to forget them from my iPhone 13. However, it turns out that the setup button had stopped working, thus I couldn’t reconnect them to my phone. I eventually gave up and bought a pair of cheap wireless headphones online but I discovered today that they are still connected to my iPad Air 4, and still function fully and are completely recognised.

Considering that, as far as I know the Bluetooth data is ICloud based, I tried to reconnect my AirPods to my phone - since I thought the iCloud based syncing would mean I wouldn’t need to use the now completely non-functioning setup button.

However, while the airpods setup screen comes up on my phone, they aren’t recognised as my airpods and I need to go through the initial setup process (obviously impossible now since I have a non-working setup button).

I just updated to IOS 18.1 in hopes that it would be a solution but alas no dice.

I am looking for a way to sync the bluetooth connection data between my iPad and iPhone, or retrieve it from iCloud or something. I am assuming (I may be wrong, I often am) that circumventing the initial setup process is possible due to the automatic syncing capabilities of airpods, everything I can find online says you only need to be signed in on the same Apple ID which I am - so why will these airpods not come up as my airpods and not require the initial setup?