This is a fairly fascinating sentence. It might easily appear in speech or fiction, probably in a really lighthearted way, but would be unlikely to show up in a journalist's account or an academic paper.
Why is this so. I'm not certain, but cases in which the usage, "I (verbed) my way" is always fine for any sort of context generally involve resistance or conflict. "He sliced his way through the whole block of cheese." "He fought his way to the top of the hill." "She argued her way into the courtroom." These could easily appear in a college paper or a newspaper.
When the verb does not clearly involve resistance, though, the use of the pronoun phrase, "his way" is a sort of usage that just wouldn't be present in formal speech or academic or formal writing. I'm not positive about this analysis, but it's very interesting.