Lando bastard book spoilers ahead:
The idea that he is a bastard is in question.
Firstly, Lyanna was probably not raped, but dug on Rhaegar.
Secondly, Rhaegar was married to Elia, so any other kids would be bastards, unless he was also married to Lyanna. Targaryens have a history of marrying more than one person, and Elia comes form Dorne, which we know is pretty groovy when it comes to things like this.
Thirdly, Howland Reed was at the tower, and is a noble, so he could vouch for Jon. The Baratheon kids were accepted as legit until Ned sent out word that they were not, which is what started the entire war. Once their legitimacy was in question, Stannis had claim, Renly got a big head, then they kill Ned so Rob had to get involved, and Greyjoy just had to be a cunt.
Forthly, we don't know what George has in mind. Jon marches on winterfell where Lyannas tomb is. We don't know what secrets it might keep. If it has artifacts that belonged to Rhaegar, it could certainly sway opinion.