You are correct that F1 chooses between the screens...and yes, multiple screens are supported, just one at a time. The error you're seeing is that your resolution is higher than KVM Remote Control supports. The max is 1600x1200 @16 bit color depth. @ 8 bit color you can go higher res. I don't know the exact limit but it's pretty high. So, lower your res or bit depth and it should work.
Here's a trick: In Real Viewer Plus you can force the bit depth down in the expert settings. This way, when you connect, the bit depth lowers and you can use KVM. When you disconnect, it raises back up. In Real Viewer Plus:
Click Advanced button
Click Expert Tab
Find Colour Level and set it to 0 or 1.
I have not tried this option so please let me know how it goes.