Hi NCubbs,
I confirm that I have the same problem with keyboard layout.
The deployment method is different from yours. I use the Google management console for View Client Deployment and automatically run View Client in kiosk mode, but the keyboard layout problem is the same.
I have a Swiss-French keyboard, which works well through the web client. If I connect locally to the Chromebook, the keyboard is also functional. By cons, as for you, it does not work in the View client, including special characters (like /*-\<>,.$àéè!....)
But it is not a US keyboard in my case, since the letters ZY are not reversed (the Swiss keyboard is a "QWERTZ" while the US is a "QWERTY"). In fact, only special characters are wrong!
Otherwise, for your problem of access to F11-F12 keys, you can use "View Menu" on the left of the screen and click on the "Keyboard" icon to display a bar with the functions in the bottom of the screen. Moreover, when this function is active, the keyboard layout also works for me !!!
Hope this helps you.