But now it doesn't matter, because I just stumbled upon a new RDP client called Remmina, thanks to this post. Remmina is quite nice, and solves both of the main issues I had with tsclient. Perhaps it will present some issues of its own, but so far, it's won me over.
I'm sure it's available in the repositories for most distributions. At the moment I'm using Ubuntu, and so to install it, I ran this in a terminal:
sudo apt-get install remmina
Here's a screenshot: