Of course you can lord over the machine and help hold the material down.
This problem happens when cutting a larger piece of hardboard as well. The endmill sometimes tries to lift the material up as it’s pulling material away from it.
I thought of adding a small wheel to the cnc machine that can press down on the material being cut. So when the endmill plunges down, there will also be a wheel that comes down on the material.
I thought it should be spring loaded and that it should be a castor so it will follow the direction of the cut.
It may not be clear from the diagrams but I need to add some 1/4-20 threaded rod. Two pieces to form the axle for the wheel and the lever axis. Another vertical in the Square hole. A spring is running from the small holes at the lower right to the hole at the top right.
I cooked this quick design up in Sketchup and will cut it out shortly and see if it works.