4of4 Surely she is found out at some point, possibly spectacularly, by exposing her dragon form to save a town or something (maybe the human form while strong isn't as strong as the dragon form). This ideally would occur after she's started to develop a reputation for herself. Also consider that the quest items might be crafted by the fairy into a gift to help her people somehow.... sorry it was so long.
3of4 Sure dragon scales are useful and all but ya know dragons molt them as they grow so they could trade with the other races. There might also be rare herbs or crystals, etc only found in the dragon territory. But do they trade? Nope, no interaction at all, they're above all that so they don't bother. But having the princess interact with them and generally being helpful to them (as she collects quest items for the fairy) she builds up a rapport and a good image for the dragons.
2of4 She sends her on quests to find super rare items scattered across the world and suggests that being an adventurer would help her in her journey. You see the fairy knows that the real problem is that the dragons are isolationists and haughty. Sure they are mighty but people fight them and manage to kill them often enough compared to their low birthrate that the population is declining. The other races fear them not just for their abilities but because they aren't understood very well.
Still stuck? 1of4 Who knows maybe this will spark something. (ignore me if it bothers you). Perhaps while getting her name from the mysterious ancestor the princess is distraught about the story (s)he tells with regard to the decline of the dragon race. She offers to help and is sent on an adventure to find a fairy. Fairies are known to help those in need so maybe, just maybe they can help. The fairy does understand the problem but realizes that words of advice aren't enough to solve it.
Are you looking for suggestions or ideas? Here's one. Perhaps because she's so advanced mentally that the dragons don't know what to do with her so they send her to a human academy for early training? Assuming she can get her human form quickly that is. Imagine the hilarity or embarrassing situations that could happen when a 1 week or 1 month old dragon enters school and interacts with her peers!