In addition to being lightweight balsa is also soft, really soft. It makes pine seem tough, that soft.
Balsa is either the softest or second softest wood known to man!
While there are many woods soft enough to mark with a fingernail just general handling of balsa can lead to dings, and very bad dents are possible from the gentlest mishandling. This alone I think rules it out.
I'm wondering if there any disadvantages I need to be aware of with using balsa (e.g. will it warp after time, etc.)?
Warping isn't an issue here.
Balsa is so soft it really can't be nailed successfully. In general glue alone is used to attach it and this could make installation tricky or very slow.
If the softness didn't put you off the major hurdle to even trying this is the very limited dimensions that balsa is commonly available in, with perhaps 36" being the longest lengths you'll easily find. So for long lengths — very desirable for a project like this — you'd be looking at an online order.
A harder hardwood1 could be the ideal material for wall slats but even common softwood would be preferable. Or, since you'll be painting, plywood. In any case if you want long lengths (e.g. a full 8') you'll either have to make them yourself or it would be a custom order.....
1 Yes, bizarrely balsa is technically a hardwood.