Yours is a very specific application of OBS Studio. It may not even be supported in your configuration. Pen display tablets are only fully aware by pen aware applications. It is unlikely that OBS Studio is pen aware. So input messages from the pen may not be recognized by OBS Studio. They are extensions of a mouse message that contain more information in it that OBS Studio may not be able to emulate.
I have a XP-Pen Artist 12 Pro
pen display monitor of which I'm sure you are aware is similar to a wacom Cintiq. It is not my primary screen in my multi monitor setup. I can display capture it just fine. Without getting into the gory details I need to mirror my primary monitor onto the XP-Pen because Adobe Cloud apps don't bring up dialog boxes and the like on the parent window. Rather, they do so on the primary monitor. Thus, I have to run Adobe apps on the primary and mirror to the XP-Pen . Of course, it is advised that both use the same aspect ratio and resolution. For coloring, I choose to sketch only on the XP-Pen and color calibrate and fill on the primary.
To mirror, you need a multi monitor Nvidia GPU. You connect the Cintiqu to one of the display ports and another monitor, the primary to a display port. Then, use nvidia control panel software to clone. With OBS, you then display capture the primary monitor.