I'd be honest with the bride, immediately, and tell her that I'd never actually worn a sari (my understaning is that wrapping and wearing a sari correctly is not a genetic skill), that I'm really uncomfortable being singled out to wear a fancy costume in the wedding when everyone else is in normal Western clothing, that I can't find a sari that works. If that doesn't work, tell the bride that she is offended at being singled out for display because she's non-white.
And I'd say the bride *is* being racist, but not maliciously and doesn't realize what she's doing. She is asking her friend to dress up in an ethnic costume she doesn't ever wear, solely because friend is a different colour. She's not asking the other three bridesmaids to dress up in the traditional outfits of their cultural heritages.
It would be different if the friend regularly wore saris, and the bride asked if she'd like to do the same as a bridesmaid.
This is actually very common in Canada/the US, and something that many people don't notice or realize. Someone who is non-white will get asked things like "where are you from?" "no really, where are you from" or asked to wear ethnic costumes at a wedding, or expected to speak their native language or have close ties to their home country, even if their ancestors have been in the country for generations, they speak English only, and they are totally Canadian (or American) culturally. Someone who is white won't run into these assumptions to the same degree, because they are assumed to be just plain Canadian.
It's not malicious, but it is subtly racist, because it assumes that someone who is say, racially Chinese or Indian isn't really Canadian in the same way that a white person is.