In the past, it was fairly common to do "border runs" - when your 90 days were up as a backpacker, you'd exit the country for a few hours, and come right back in. I met many people doing this in most South American countries.
In the past, the common way would be to do a border run. Head over from Foz Iguazu to Iguazu, spend the day checking out the falls, maybe spend a night, and then re-enter. That's 90 + say, 89 days (one in Argentina), and 179 days is still more than 5 months :)
However, as you've mentioned, the tourist visa has a rule now that you can only be in Brazil for 90 days out of 180. So it's quite clear, if you want to be a legal visitor, without overstaying, the only legal way you can manage this is to get a non-tourist visa.
Bear in mind, as well, if you're on a tourist visa and interning, and they don't allow that on a tourist visa, it's definitely in your best interests to get a work or student visa. Just because it's time-consuming is NOT a valid excuse for breaking the law, and if officials work out what you've been doing, you're going to have a tough time explaining.