There are lots of things to consider making each case different, however I would choose the DL1000 over the KLR650 for two up touring in South America.

One big factor to keep in mind is the comfort of the passenger as an unhappy passenger makes for a lot of stops and avoiding of roads that are rough on them. The DL is a much better bike for this as it has a bigger cockpit, decent seat, etc. It can of course be done on a KLR but it's a different experience. A custom seat on the KLR would be a minimum requirement if you ended up going in that direction.

I have previously ridden around Turkey two up with a pillion on on DL1000 and just did a short bit of two up riding in South America over New Years on a KLR. Most riders and passengers will find the experience on a DL more comfortable.

In regards to roads, the main limiting factor will be the fact that you are two up, not which bike you will be on. At minimum, put a knobby tire on the front of the DL and it will be fine for the vast majority of roads that you will encounter on the the standard routes. If you want to be doubly sure, bring over front and back Heindeau K60s and you'll be fine unless you really get off the beaten track or encounter mud. If that happens, then the main issues of two riders and all their gear will make any bike very heavy and you will quickly learn which roads to avoid in the future.

Happy planning.