Your argument is basically Kurepa's proof from his thesis [Ensembles ordonnées et ramifiés][1], see page 96 (a footnote has Aronszajn's construction).
As noted in the comments you need to show that what you suggest in step 4 is actually possible. Kurepa has already investigated the tree of well-ordered subsets of $\mathbb{Q}$ in some depth and he could indeed `simply' assert that what you suggest is possible.


  [1]: https://eudml.org/doc/192853