It didn't look like HP had any software related to the touchscreen I am using (HP L6015tm) so tried installing Elo's control panel software to see if maybe I could turn on their ForceMouse mode. Unfortunately, I couldn't apply any settings once inside the application.
At this point, I understood that I essentially needed the touchscreen to act like a trackpad. I wondered if a trackpad driver could somehow replace the generic mouse driver for the touchscreen.
I eventually came across a few comments about software called Tablet Pro STUDIO which I believe is an evolution of software that was originally called touchmousepointer. The links to touchmousepointer were all dead, but after some digging I was able to find what I believe was the final version "touchzoomdesktop2105".
I am happy to say that things are working! 

In summary (in case someone stumbles upon this in another 16 years)
MAME:Input Device Options:
- Lightgun Device Assignment: Mouse
- Lightgun Input Provider > win32
Advanced Options:
Install TouchZoomDesktop Component (touchzoomdesktop2105)
Settings > Pinch > Mode > Touch to Mouse Mode
I haven't played around with the other settings in TouchZoomDesktop yet. I am excited that I can move forward with a new bartop touchscreen build!