I think I would agree with the other ladies. She is still very young, and maybe bringing her to the pool outside of lessons and saving that for fall or next summer would be better. Or, you can bring her to the pool, and let her make up her mind to go in with you and the baby in the deeper areas, for instance. Give her the control. I would be very sympathetic to her fear -- say, I know you are afraid, why are you afraid, do you know why? (in case she can't articulate it). Then, have her hanging around kids her age or older who are in the deep pool, having loads of fun. My little guy is three, and also hates getting his face wet and having his hair washed, so we have little issues with the pool as well. We have him "swim" in the bathtub, practicing putting his face in the water. Seeing bigger kids and having them talk to him about swimming and putting their face in, etc., and play in the water, was a big help for him. I've found that, with him, talking about things away from the event, a lot, is helpful. We have lots of discussions in the car.
I think your instructor's advice is good, if it is important to you that she learn this summer. However, be prepared for it to be traumatic for you and her, I think. Kids, when they dig in their heels, dig in their heels.
If you don't have a pool in the backyard or a concern like that, I'm not sure I understand why it has to be this summer. I understand that you had a traumatic event, but there are statistics out there that show even kids who know how to swim, can still fall in, panic, and have trauma. With further passage of time, she might not be so adamant about refusing to go in. And I would still try taking her to a pool and presenting her with the ability to go in with you on her terms. (You have the one year old in with you, and poor daughter can't come in because she's scared, etc.) More of a gentle work around the issue, giving her some control, rather then forcing it. And maybe have someone other then you take her to the pool. Sometimes kids are very different with others, parents do have a different dynamic, which may be why the instructor suggests that you go away during the lessons. Your daughter probably senses your anxiety on her behalf, which may verify her fear, despite what you are telling her. Kids pick up on our body language and tone, particularly girls.
Good luck!