This story is old enough to shave. I recently have heard in posts and in private messages, this exact problem. I have noticed from long distance bicycle rides (RAGBRAI, criss crossing Iowa in summer @500mi. etc) that when I start out, I am super hydrated with a sports drink like Gatorade, only better. I just don't sweat, I stop about every ten minutes to pee, but I am dry as a bone sweatwise. By the middle of the second day, the flood gates have opened I am carrying extra water in case sweat gets past my two sweatbands into my eyes. Like a spitting cobra had just nailed me, it burns like saturated amonia solution. Yeah its got salt. Salt builds under anything like my ring or watch. I get a daily buildup of something resembling putty under my watch. It's salt putty.
So, here is my thinking, before you take your child for a sweat test, train her to sweat. Start by hydrating her on say, Gatorade, maybe sneak a little salt as suggested into the drinks. Then dress her up in winter clothes and play aggessively with her until she is shot. Do this several times a day, you know when to stop, every day until she sweats like there is no tomorrow. My bother, if asked what he is, would probably answer, he is an athelete. He has been one his entire life and at 67 he still starts his daily workout with 200 situps. He clued me in decades ago about training yourself to sweat. I am the ultimate empiricist and did my own sweat tests to confirm his belief.
Having said this, there are many "preventors" that can interfere with this plan. If your daughter is taking medications that cause dry mouth, like anti-siezure drugs, anti-depressents, or whatever. A talk with her Pediatric doctor will give you that answer. Since she has bouts of diahrrea, pedialite, if she needs some contains loads of electrolytes. You know the drill, this is not something you do without running it by her doctor.
Toronto Childrens that Printer mentioned, is a freely given, hand polished pearl of wisdom. Although I subscribe to the axiom, trust but verify, nothing he has posted, that I have seen, is anything short of hard won, well paid for, wisdom. Just think everything he says is being shouted, he says little and speaks volume. I am somewhat guilty of the converse.
Call your family doctor, have him order a genetic screening for CF and though I have been chided for this recommendation, go see a GOOD hopefully specializing in CF, Genetic Counsellor. I agree with everyone, this doctor may be attempting to remain in the good graces of your health Insurance, by avoiding an expensive test. It isn't anymor expensive than an MRI, but it isn't a CBC either.
It's a thought.