Dropbox would be fine. You could use a combination of get file, save file, delete file to perform a move. You just need to ensure consistency of naming or appropriate parameters being passed through - so not a share sheet option in Dropbox, more a direct action.
The iSH file provider isn’t going to be accessible without the manual select file and navigate step.
That would still require Shortcuts to do some moving for Dropbox and at that point, why not just do it with Shortcuts?*
iSH, might show up for Hazel on the Mac I guess, and you could trigger that by a file move within iSH.
* The exception I can think of for this is with large files, but you might want to run that on the Mac say via an SSH session to save mobile bandwidth.