Try not to use cool/cold water. Warm bath/washer & let the water dry on her skin. Cold/cool can restrict the blood vessels & push the temp higher.
I'd try another at least half dose of nurofen unless you can get the warm washer to work.
Works well for me with DD.

And just remember ANY fluid is better than none. Dd was given red cordial in a syringe last time I took her to er vomiting. Most of it came back up, but the little that stayed in was all she needed. They absorb it through the mouth too, so they still get it when some comes up.

Keep the paed appointment. Even if you have no results. Anything that they think needs doing will be done. Be honest & tell them your concerns. Everything. Dd1 had her tonsils/adenoids etc removed at 5. She'd never had tonsillitis, but she was an open mouth breather, who snored & always had black rings under her eyes. Just from the enlarged tonsils/adenoids.

I would suggest a 2nd opinion, but hopefully between the paed & ENT you'll work it out & won't need to worry.

Dd3 is almost 4 & I'd be shocked if she chose to put herself to bed at any time! Even sick

You know your dd. I had bloods done on my 8 week old dd2 for recurrent chest infections. My gp decided the paed was the answer. This is how we found her heart murmer, among other things. If it hadn't been for being able to have faith in my gp, we'd have never thought to go further I don't think.