I too am struggling to reason out my reaction, which is No Way.
It's not the chemicals. It's the first hint of self-awareness, the interest in appearance, however fun or creative or "dress-up", in a 5yo that seems a few years too soon... there is so much pressure on girls to look a certain way and I am not suggesting for a minute that the OP's dd is a budding beauty queen or that OP is playing dolly; it's the tiniest, tiniest peep over the slippery slope of interest in appearance - to the "everyone else is allowed", to nail polish at the school disco, to make-up, to becoming overly interested at too young an age in looks and body image and appearance, to anxiety over those things and to mental health issues that affect so many young women and girls in today's society.
I know this will come across as a drastic and ridiculous leap from "bit of pink hair dye fun" to "anxiety over body image" but it's the closest I can get to why I would say no - because I would, and do, want my DDs to enjoy the freedom of not giving a damn what they look like, because once they do start caring about the way they look... they'll do so for the rest of their lives. And I want to keep them back, even from the tiniest, tiniest peep over the slippery slope, as long as I can. (That's not to say I won't be lending them the straighteners or banning makeup when they're old enough though. Going too far the other way is just as bad!)
Ugh. Sorry for inarticulated, clumsy and overbearing argument.