Trouble Updating Apple Watch 6 OS (os8.8.1) with iPhone 7S (iOS 15.8)

I have an Apple Watch Series 6 running watchOS 8.8.1 and an iPhone 7S running iOS 15.8 (that is as high as the 7S will go - it cannot run iOS 16 or higher).

When I try to update the apple watch from the iPhone app, it tells me to update my iPhone software, but the iOS is already as up-to-date as it can be.

When I try to update the apple watch from the watch itself, I get the following error:

Unable to Check for Update - an error occurred while checking for software update. (cancel/try again).

I am connected to WiFi, the watch has more than sufficient battery charge, and I've already tried unpairing & re-pairing the watch.

How can I solve this issue?