Well there is a fabric gate but I would worry if you had one that chewed.
Walmart.com: Evenflo - Crosstown Soft & Wide Pressure Mount Gate: Health & Safety
This would be the one I would try for a climber, it's clear plastic, nothing for them to grab a hold of to climb. It can even be mounted so that it swings and opens with the handle if need be.
I have a similar with the same setup, just different paneling with the standard holes and I love how easy it makes it to use.
I have one climber and my gate doesn't quite make the cut but it does deter her some. I would have loved to have gotten the gate I listed below but I didn't find it until after I already bought my gate.
Amazon.com: The First Years All Clear Swing Gate:...