What makes you think she is not coeliac? I have a coeliac DC who had many symptoms, including the ones you're describing (severe lethargy, irritability). I totally understand that it's horrible feeding a child food that makes them ill (been there, done that). And you'll have to do it for six weeks before the blood test, which could be awful. Is she having any other symptoms like bloating, tummy aches, yellow floaty poos, wind, "glazing over" (a kind of fazing out, staring into space thing which lasts for a few minutes at a time)
But for a coeliac, it's critical to get diagnosed - for example, it's not just wheat that coeliacs need to cut out (barley and rye, and non-gf oats). Also, they need to be much, much more careful about small amounts of gluten, including cross-contamination than do people who are 'just' intolerant. And also, they are entitled to further health monitoring, which significantly helps with keeping the diet to the required standard (annual blood tests to check for antibodies etc, gf food on prescription etc.)
I second the person who suggested a gene test (HLA-typing blood test). I wouldn't stop the diet now, since it would be awful to have to restart again a few weeks later. Instead, I'd go ahead and request it straight away whilst continuing the gluten-heavy diet. If the results come back negative, you don't have to continue. But if they are positive for the DQ2 or DQ8 "coeliac" genes, then you can feel confident in what you're doing (though note that about a third of the population have those genes, though only a smaller proportion of them will be or become coeliac).
By the way, I think your husband should be tested for coeliac too.