I think you've handled it well, Lemon.
I was given an old lady name by my parents in the 70s. I'm named after my grandmother, who was born at the turn of last century.
Apparently my parents really deliberated over it, and rationalised to each other that I could always opt for my middle name if I really hated it.
Then, just after I was born, a load of them turned up in the obituaries, and they were really worried they'd made a terrible mistake.
Long story (slightly) short(ish), I hated my name from about aged 10 to early 20s. Not a soul my age had it, I couldn't get any plastic tat with it on, and everyone of my parents' generation trotted out the 'necessary on a bike line' at me, like they were some kind of original comedian and which I don't even get until I was well into my teens.
I love my name now. I won't say I can't believe I used to hate it, because I remember what it's like to be a teen and just wanting a name from the era I was born in.
Rose is an elegant, timeless, classic name and guaranteed she will grow to really like it once she gets passed these awkward years.
Humour her, and she will get past it much more quickly than if you refuse to indulge her.