How about this...

Use the tablet as a virtual touchpad. That's the way it is by
default (nothing new here). With the big picture on your big
monitor. That's the way it currently is.

But...

Have a single place/button on the tablet screen you can tap to
toggle between touchpad and keyboard. When you want to use the
keyboard, it pops up to cover the whole tablet screen (still in
landscape mode). The main difference is that the keyboard fills
the whole tablet screen and doesn't show up on your monitor.

Either that, or have both the keyboard and touchpad showing on the
tablet screen. But I suppose you can get away with temporarily
using the keyboard, the way it is now.