Lincoln is an ambitious and slippery character. He always attached himself to whoever seemed to have the upper hand at the time to advance himself -- an Edwardian until Edward IV's death (gaining lands and titles) and a Ricardian until Richard III's death (becoming Richard's heir after Middleham's death), he submitted to Henry VII after Bosworth, but then almost immediately began conspiring against the new king when he gained nothing under the new Tudor regime. He supported the Lambert Simnel farce in OTL, but in ATL he has no reason to hide behind an imposter pretender and he can just declare himself the "true" Yorkist heir.
It could be argued that Titulus Regius is invalid because it was passed under duress. And if we're going to look at the technicalities, then it should be noted that George's line is not out. The attainder deprives his line of the lands and titles bestowed upon him by the crown, but does not exclude it from the crown itself. (Indeed, Richard of York's claim to the throne stems from the idea that the line of succession cannot be changed by parliament in any event -- if it could, then the Yorkists were never the rightful kings, as parliament picked Lancaster in 1399.) But in any event, this is not a time for legalities. The fact is no one can agree means that it can only be settled by war.