You'll see an iOS 18.7 update, but that's not the one you want to select. Instead, just below that tap on Upgrade to iOS 26 ...

Plug in your phone and make sure you're connected to WiFi. (The download will require power and a good connection.) Go into your Settings > General, then click on Software Update. From there, choose ...

The latest PS5 system update will finally allow your DualSense controller to pair across multiple devices without having to de- and re-pair every time you switch.