I eventually found a solution to this via this blog post by Falafel Softare.
- Download Microsoft's souped-up RDP client called Remote Desktop Connection Manager 2.2 (edit: no longer available so try your luck with Remote Desktop Connection Manager 2.7 - see notes below)
- When you add a new server, go to the 'Remote Desktop Settings' tab of the Properties and select 'Same as client area'
- Remote Desktop Connection Manager will then display the remote machine using the client machines DPI settings.
I've written this up in a bit more detail with screenshots on my blog.
edit: Sounds like this might only work with Windows 8 clients, getting reports it doesn't work the same way with Windows 7 clients.
edit: Microsoft is no longer offering V2.2, only V2.7. Sounds like it will still work if you tweak the settings: Browse to the location of RDCMan.exe. (C:\Program Files (x86)\Microsoft\Remote Desktop Connection Manager by default). Right click RDCMan.exe select properties, select Compatibility tab and uncheck "Disable display scaling on high DPI settings"