We eventually settled for Rio Mutum. On the map it looks close to Cuiaba and we were afraid there would be many people, but the area is quite empty. We camped on someone's garden - google maps show a campsite there, which seems to be a mistake upon which the family capitalized and started offering the camping ... But it's a nice place on the river and we had it to ourselves. We paddled only up and down the river - going up, open water ends after 3 kms, going down it goes forever but the current is not negligible and coming back was demanding. Also interesting is that overnight the water level dropped a bit, but the current increased. This area doesn't seem to have jaguars (as people camp here freely), but the river has Giant Otters, which we saw several times. Pretty good.
Honestly, this is probably it. There is far less open water in Pantanal than we thought, so it's only rivers - and even rivers are often overgrown and impassable. We enjoyed the two occasions where we could boat - Rio Claro and Mutum - but overall this idea is less feasible than we thought, between the lack of open water, difficulty of access, difficulty of getting safer campsites, heat (35 deg usually) and currents.