Sometimes you need to wait a bit for the RuneScape client to load before "binding it".
If that didnt help, try the below steps.
Uninstall RuneMate, and check if theres still files at the following locations:
1. C:\Users\<your username>\RuneMate
2. C:\Program Files (x86)\RuneMate
If any of these folder still contain files, delete all of them.
Then simply redownload and reinstall RuneMate from
Download | RuneMate
RuneMate will come with its own verified tested working version of Java, and that should fix your issue.