I think your best bet is to SSH in yourself and do updates and commits from the command-line.
First off, I'm not aware of a GUI SVN tool that can act as a front end for a remote working copy. Most GUI tools use Subversion libraries to implement their functionality, since invoking a command-line process and parsing the results is such a pain and really slow. Thus, such a GUI tool would have to either (1) utilize the SVN libraries installed on the remote machine (which would be a nightmare, if not impossible) or (2) execute commands over SSH on the remote machine, which (as mentioned above) would introduce a parallel, less efficient way of doing things.
The other alternative you haven't mentioned is to use VNC or some remote desktop tool to manipulate a GUI program running on the server. Since you use TortoiseSVN, obviously you're on a Windows client, but you haven't said what the server is. If you control the testing server and are able to set up VNC or something similar, this may be worth trying. Who knows, the lag from VNC may be less that doing SVN operations on a WebDAV mount.