Not sure I get this. I don't think Dany and Tyrion will oppose Jon, if anything, his parentage will cement the alliance between Stark and Targaryen. In fact, I wouldn't be surprised if it caused other houses to swing to support him.
Jamie is bringing news of Cersei's treachery, so they know what's coming from the south. Now we've all forgotten Dorne since the Sand Snakes were killed/captured, but they traditionally support Targaryens, right? Jon's claim to the throne is quite strong at this point, add his righteous reputation, and who left will fail to support him? I suspect that even Dany will realize he's the best king, because he doesn't want to be.