I think it all depends on how much snow you have to go through. Normally, there's very little on the Puna, so I think you should be fine with your lighter shoes.
The only two peaks I felt a bit cold on my feet was on El Muerto and Pissis East. On those two peaks there's a lot of snow and even some glacial sections.
I should add; the route I took on El Muerto. We climbed the glacier and there's obviously a scree route on the Chilean side.