AirPods 4 beeping (Find My chirp) but won't show up in Find My app – Mac only user

Hey everyone, I’m running into a frustrating issue with my new AirPods 4.

They keep making the "Find My" chirping/tracking sound randomly. I know this is an anti-stalking/separation alert, but the problem is that they do not appear in my Find My app or my iCloud device list at all.

My Setup:

• I do NOT have an iPhone. I am using a Mac as my primary device.
• The AirPods are paired via Bluetooth and audio works perfectly.

What I’ve already tried:

• Factory resetting (Double-tap + Long-press on the front of the case until the light flashes amber).
• Forgetting the device in Bluetooth settings and re-pairing.
• Checking System Settings > Bluetooth > (i) — the "Find My Network" toggle is there, but turning it on/off doesn't make them appear in the Find My app.
• Location Services are ON for Find My on my Mac.

They seem to be in a "limbo" state where they aren't "claimed" by my Apple ID, so they think they are lost/tracking someone. Has anyone managed to get AirPods 4 to officially register with an Apple ID using only a Mac?

Any help is appreciated!