You don’t have to assume any unofficial races. Russia holds an international tank biathlon every year. There are plenty of videos online.
There used to be a lot of them going up Mount Washington in Pittsburgh, some for cargo and some for people, prior to the Liberty, Fort Pitt, and Wabash tunnels being bored through to facilitate transportation. I’m sure some of the operators raced each other. And since the Duquesne and Monongahela Inclines still exist, I’m sure if the owners had the inclination they could set something up.
I’m assuming they were distinct funicular lines and not cars connected on the same line (which I think was the joke @Shalmanese was making).