Cannot pair factory reset series 3 watch to iPhone using iOS 15.4.1

I have a factory reset Apple Watch series 3 trying to pair to a iPhone 13 using iOS 15.4.1. The watch has been charging for ~3 hours, so I have to assume that it is greater that 50% battery level, as I cannot see the level on the watch face.

I can start the pairing process. I get to a screen on my phone that says:

Software Update

A software update is available for this Apple Watch. Update now and continue setting up.

I click Update Now, and the screen goes to 'Checking for Update..."

It then comes back saying:

To update the software on your Apple Watch, you first need to update the software on your iPhone.

Go to Software Update settings.

I click the "Go to" text, and the screen switches to Settings->General->Software Update. The screen says:

iOS15.4.1

iOS is up to date

This is where I get stuck in a loop. I go back to the Watch app, and it still says my phone is out of date.

What should I do?