It's a "pick your battles" kinda thing, you're not gonna get someone at Starbucks write your name down properly if it isn't something they can easily hear over the murmur of a crowd.
That said, if I were in that situation I would double down on it in anything formal - documents, graduation ceremonies, weddings, lawsuits, etc.
I have a coworker named Domas. It's pronounceable enough, but he uses Thomas (which is actually just a variation of the same name in some cultures) as his Starbucks name for...reasons.
That said, if I were in that situation I would double down on it in anything formal - documents, graduation ceremonies, weddings, lawsuits, etc.