I've biked in both Pucon & Bariloche, but I was touring on my own bike (it's a wonderful trip from Puerto Montt, Chile, to Bariloche, Argentina).. I'd say it's likely that you can rent bikes in both towns, because each town has lots of outfitters for outdoor activities. In both places, the number of paved roads in minimal, so I imagine it would be mountain biking you'd find yourselves doing.

In Pucon, I did an excursion with an outfitter to climb Villarica volcano. I have no idea if they're still allowing it, since the volcano is not dormant. We climbed up thru the snow to the rim of the volcano. It was amazing to look into the caldera and see the mini-eruptions of red-hot lava. The fumes there can be deadly, so you cannot stay around the caldera for very long. In restrospect, it was probably pretty dangerous. A couple of the mini-eruptions came halfway up to where I was standing looking down into the volcano.