I think people are missing the point. It's not about being cross with a 6 year old, let's face it the 6 year old has almost nothing to do with it - she's just wearing what her parents told her to wear.
That dress is a bridesmaid dress. If you go to John Lewis, Next or Monsoon those dresses are usually slightly separate from the other girls dresses because they are easily identifiable as bridesmaid dresses. That's not to say girls don't wear them to other events, but they are bridesmaid dresses none the less.
If the OP asks the bride and the bride is fine with it then there is no issue. Otherwise she risks offending the bride, not because the bride is a bridezilla but because there may be other issues that the OP is unaware of.
For example, I know a bride who has 7 nieces and choose not to have them as flower girls /bridesmaids because there were so many of them and in addition to her 3 sisters the bridal party would have been massive. She couldn't choose one or two without upsetting the others so none were. If a girl at that wedding turned up in a beautiful bridesmaid dress it would put the bride in an uncomfortable position with her nieces who could understandably be upset by it.
I've only been to one wedding where a girl (approx 5yrs) came in an ivory flower girl dress, the bride said nothing negative at the time but has commented since that people who see her photos ask her who the flower girl was. She is a little embarrassed to say she doesn't really know, it was the little girl of one of the grooms colleagues girlfriend. People just assume that girl in bridesmaid dress at a wedding = a bridesmaid.
At the wedding, people told the girl and her mum how beautiful she looked (she did) and what a pretty dress it was (it was lovely) but privately they were rolling their eyes.
I heard comments from other guests, most appeared to think that it was a weird thing to do, most assumed that the mother was trying to shoehorn the little girl into the bridal party.
Unless the bride was asked I think it shows thoughtlessness for the couple at best and attention-seeking behaviour (from the parents, not the child) at worst.
If the bride is fine with it, go for it but at least give her the chance to say no.