I totally understand about picking your battles but honestly this one I would pick. I can let the outfit slide, sox don't match no big, vegies raw but not cooked fine, but personal hygene is a different story.
I have found that with a little work you can "fight the battle" respectfully, here are my stages:
I would give her one chance to talk about why she isn't/dosen't want to wash properly. If that dosen't get to the issue then I would offer to help, ie we can cut a few inches, pick a new fun shampoo etc. Then ultimately the hair would get washed as often as I think appropriate. Here my dd's shower nightly before bed - my 10yo washes hair every other evening and my 12yo every evening. Nothing else happens until the shower, no dessert, TV shows etc.
If the hair dosen't get done properly its back in and mom does it, yep

My dd's are 10 and 12 and it only took me re-washing it once to make the point although my 10yo sometimes still asks for my help rinsing.
(fwiw I did it with teeth as well and it was not fun re-brushing my then 8yo's teeth but again I only had to do it once
Another thing that may help is that at around age 9 I found that the kiddie shampoos didn't get the hair clean anymore. However some of the adult shampoos are too difficult for them to rinse out so either way the hair still looked yucky after a shower. We finally found a suave with a light conditioner that works really well for them.
Good luck - I know its easy to say "you should" but I hope you figure out what's going to work for her and for you.
I want to re - recommend:
The Care And Keeping Of You
from the American Girl series of books.
If my dd's read a tip or some advice in a book or magazine its good advice - vs when they hear the same thing from mom its like the charlie brown cartoon wa wah wa wha wah wa aha
TJ