Update WebView without using Google Play
If you don't mind factory resetting your phone, you can try to unlock your phone's bootloader, flash TWRP Recovery and then install Open GApps, which installs/reinstalls Google apps and can give you ones that you cannot get from Google play.
Disclaimer:- I am not responsible for what may happen to your device during this process.
It installs WebView with it (if selected) and can remove stock (AOSP) Apps (Your default WebView). If it does not work you can still flash SuperSU and uninstall the system app.
-Daniel ✓

Edit: Since you already have full root access, you can try to uninstall WebView and install a fresh one from the web. Just be sure to download one prior to uninstalling.