We had exactly this problem and I posted up a very similar post a while ago and got loads of good advice as well. I love all the ideas, as you've said, the great thing about this site...
We chose our doll together, but it was one of those disney dolls and i have to say once we got her clothes off to go in the bath my dd freaked out - well they are
pretty weird! We tried everything, including those jug things, and the visor, which didn't work for us (but may for you). The thing I found worked in the end was based on one of the bits of advice I got on here - its important to remember that there are a lot of natural oils in hair, especially children's hair. And so most children won't actually need shampoos etc. that often - our dd has long thick hair. She went through the whole phase of refusing the bath even, but I gently reintroduced and took things really easy, then I would sing as we washed her with a flannel (I now sing as she washes herself, and I pretend to wash my own corresponding body parts), actually a Tracey tip - 'this is the way we wash our arms, wash our arms, wash our arms etc. etc.', then i'd get to her back and I would skim the back of her hair. I started doing this for a while and then added 'wash our head', then 'wash our hair' but always only with a flannel so her hair would eventually be totally wet, but none was going in her eyes. I realised that I didn't need shampoo at all. Now only on very rare occasions (when there is yogurt and sand and who knows what mangled in it) do I use shampoo and then only a very very very tiny bit, I rub it between my hands first so its coating them and then rub it over her head. It then can be wiped off with several goes of the flannel and no chance of it going in her eyes.
Thats what worked for us anyway, I'm sure you'll find the magic formula for you! Good luck, I do know how stressful it can feel