Can I force Steam to install games on a drive that isn't C when in Big Picture mode?

Yes and no. While Steam doesn't support this directly that I know of (though I rarely use Big Picture mode), you can use a symbolic link to make Windows "lie" to Steam. This will require a keyboard and/or mouse for the initial setup, but shouldn't require them after that.

Assuming your SSD's Steam install is at C:\Program Files (x86)\Steam, and you want your games to be stored at F:\SteamApps:

1. Make sure there's not already a folder at F:\SteamApps (or choose a different path).
2. Move the folder C:\Program Files (x86)\Steam\SteamApps to F:\SteamApps. At this point, no SteamApps folder should exist in your SSD's Steam folder.
3. Open a command prompt and run mklink /J "C:\Program Files (x86)\Steam\SteamApps" "F:\SteamApps"

Afterwards, if you look in C:\Program Files (x86)\Steam, you should see a shortcut leading to your HDD's SteamApps folder. When Steam queries for this folder, Windows will seamlessly send it through the symbolic link.

You can be more specific with your symlink, and only move certain games over, or everything under SteamApps\common, etc., but for the purposes of Big Picture mode, this should be your one-time-only setup step.

So the answer is sort of. You can't choose an install location or set a default location for big picture mode (at the moment). There is no configuration for this, it will always use the default install location which is the drive Steam is installed on.

And that's the key. You can move your steam installation to a different disk and it will default to that location instead of to the one on your SSD.