Stay with the handgun and be more situationally aware.
I have done long distance cycling in the Canadian and US rockies. Up in some VERY remote areas. I remember pedaling over whitefish Pass in an are where the road was cut into a hillside. There was a 6 to 8 foot shelf immediately to my left and I wondered how many cyclists had been stalked along areas like that before.
Big cats can get aggressive with humans but they usually just run away. That said, my friend Bonnie had a mountain stare her down on Galton Pass in B.C. a few years back.
ETA... A few years back, me, my son and a friend were in deep B.C. We'd already pedaled over Lodgepole Pass and were near the top of Cabin Pass. It was windy, hot and dry. We'd put in 70 miles so far in the day and were pretty quiet and drained. Right near the top of the pass, a mountain lion jumped up out of a large clump of grass and ran away. We barely had time to get out bear spray out of our holsters. I carry my bear spray on a custom holster on my handlebar near my right hand.