My iPhone won’t let me get a new update for my new Apple Watch

Try this step:

Go Setting --> Click General --> iPhone Storage --> Scroll down and see whether there is an existing iOS 15.7

If yes, please delete it and head back to software update and download again

Can you show what's the error?

Hi. That's your existing system file. Find iOS 15.7 update file

Update your phone to ios16 first,it's at the bottom of the page in your screenshot.