Hi Glimmer,
You can certainly rent a car in Cuiabá and bird the Transpantaneira productively, although during the dry season the dust can be an annoyance. The best birding is off the "highway" on the private properties of lodges like Pousada Piuval, Pouso Alegre, Pousada Rio Claro, Araras Lodge, Fazenda Santa Tereza (now Southwild Pantanal) and the Hotel Mato Grosso at the bridge over the Pixaim river.
There is a useful article by James Loewen and Giuliano Bernardon (of Birding Pantanal) in Neotropical Birding volume 6, covering the Transpantaneira. Definitely worth tracking down!
With respect to Porto Jofre, there aren't many accommodation options there. The village consists of the large Hotel Pantanal Norte (mostly known simply as Hotel Porto Jofre), two more recently constructed camps, and an older campground/pousada, in addition to two or three fishermens' houses. As such I wouldn't recommend trying to show up without a reservation, as these places are likely to be packed during the high season (July to the end of September). Some of the lodges along the Transpantaneira (including Piuval and Araras Lodge) offer day trips to look for jaguars, though you wouldn't have as much time on the river as you would if you stayed at Porto Jofre itself.
Reliable operators include Birding Pantanal, Pantanal Trackers, Pantanal Bird Club, and Biodiverse Brazil Tours. There are other good companies out there too, this was just a short list off the top of my head.