Get Public Beta on Apple Watch Paired to Family Member's iPhone

I don't have an iPhone, but got an Apple Watch SE last year. I want to enroll my Apple Watch in the Apple Public Beta Program. My iPad and Mac both are receiving beta updates, but the instructions for enrolling an Apple Watch sound like they require an iPhone. Is there any possible way to enroll my Watch in the Public Beta Program without installing it on my mom's phone/watch or anything?

Apple Public Beta Website: https://beta.apple.com

Thank you!