This is a classic problem in off-site storage locations. Many, many people (including many professional offsite storage warehouses) store in cardboard boxes without issue. However, many small wine coolers have an issue with humidity control. I'm not familiar with Climadiff but one of my cabinets of a similar size has a persistent mold problem near the condensation collection area. That could quickly get out of hand with cardboard in the cooler.
The best solution that I've heard of (short of sourcing additional shelves for your unit) is to cut to length sections of PVC pipe with a diameter large enough to fit a wine bottle inside the tube and stack them inside your cooler. They will be very stable and inert so won't smell, get moldy, etc. and it makes accessing any particular bottle very easy. It will, however, seriously reduce your total storage capacity (as you've noted with the cardboard dividers).
The only other issue that I would point out is that with any similar solution you'll likely want to leave some space at the front or back of the cooler (don't cut the tubes to the exact depth of the cooler) to allow for air circulation. Without proper circulation, temps in the cooler can diverge significantly from top to bottom.
Eric
