An adaption to the binder clips would be to put some adhesive putty on either side of the clip to hold it in place.
If you put something like Blue Tac on either side then this will give it more to grip onto and it should keep it in place for your purpose!
There are always larger clamps and clips that you could buy in order to replace what you already have but in terms of working with what you have this solution is a goer.
A second possible solution which may or may not work is to stretch a few elastic bands over the side of the monitor (keep it on the plastic edges in order to keep screen un-obscured) this would probably work better if you had something underneath the bands in order to make them rise away from the screen and give it more of a pushing force to push the screen together, but again this may or may not work, worth a try though!
Thirdly and perhaps the most 'hacky' method would be to improvise your own binder / clip. Get two sturdy bits of wood / metal / plastic, either as a rod or thin block type shape.
Three quarters of the way up the two rods / blocks (hereinafter referred to as 'the rods') place a wedge of paper (at least as thick, if not thicker, as the monitor) and then wrap an alastic band or two around the two rods above the wedge of paper (one loop below in order to keep the wedge in place) and wrap a few times until fairly tight and you can use them sort of like chopsticks.
You can then use them as you would a normal clip and add Blue Tac or something to the ends to make them more sticky if need be! Below is an image of chopsticks which is basically what I am trying to explain:
Obviously you'll need to to a bigger scale with a larger wedge in the middle but this is the premise I am aiming for, a couple of them may be better than one on its own.
Here is a link to how you can make these chopsticks on YouTube.