Your argument is basically Kurepa's proof from his thesis Ensembles ordonnées et ramifiés, 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.

Your argument is basically Kurepa's proof from his thesis Ensembles ordonnées et ramifiés, 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 had 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.