I made a first draft of the script using OnGetNewName.
MoveToBaseTogether.zip (1.9 KB)
It seems that I won't be able to use it with the Advanced Rename dialog, because the script has side-effects (moving related files) and these are triggered by simple opening the AR dialog or previewing the new names.
The same old warning in this thread about side-effects still applies.
The alternative is to move the way of defining the new name to the script and use it in a button. I think that is a bit more tiresome, as keyword codes can't be used there and I can't quickly apply a one-time substitution. Not pretty, but can be done.
Ideally, the rename presets would be able to save the user defined paths and filters and the side-effects would not kick unless the user decided to apply the rename.