Once detected by Jagex, it doesnt matter if it was a different IP.
The thing is that a Private IP, is less likely to be flagged and therefor your account less likely to get banned.
Login using the regular client, then using RuneMate.
If ban occurs at regular client, its delayed.
If it's RuneMate, shoot Arbiter a message asap or post here.
As he mentioned in his post, he applied a couple of changes.
We can not share this information, as Jagex might find out about any vulnerabilities.
The less we share, the more we care <3