Trust your intstincts. It is most probably a contact dermatitis as a result of the carpet. Carpet is a mite, dander, and allergen haven. Not only that; but the carpet fibers, themselves, contain harsh/synthetic materials that also contribute to eczema flareups. Don't discount the fact that your daughter's body has proven that it is capable of healing itself of eczema simply because she has experienced a new flareup. I don't believe that this is a remission flareup (or a flareup that came as a result of suppression). It appears that your daughter's skin is demonstrating a new flareup, as a result of her new allergen/irritant contact. In essence, her body (her immune system) is functioning properly. It alerted you immediately of the toxic culprit so that you could respond accordingly. This is why you were able to make the quick connection (or identification) of the allergen/irritant culprit.
As a result, the bigger question that you will need to answer or address is what to do about the carpet. You, along with your homeopathic doctor, have obviously proven success in restoring your daughter's internal balance to a point where healing, as opposed to suppression, is taking place (the homeopathic doctor is to be commended for not pushing more medicines/supplements in order to allow the body time to heal). So, in your case, it's not really an issue of what new medicines or alternatives you will need to try; it's simply a case of removing the allergen/irritant culprit. When that is done, the skin can complete the healing process it started when it shouted with a flareup; as a result of the toxic contact.
The beauty in reading the posts between you and John is that people can actually begin to see, and hopefully gain understanding, in how to read and respond to the body's message of an eczema symptom. The both of you are so "dead on" in your analysis; yet, I see the two of you lose confidence in what you have learned and successfully achieved when a new flareup arises. Anybody can develope an eczema flareup as a result of toxic contact. Keeping that in mind, and understanding that there is an array of allergen/irritant culprits that could cross the human path, don't let a new flareup (especially as a result of a new contact) shatter all that you've learned and achieved.
Case in point: my middle niece had been eczema free for four years; yet, she experienced a total body eczema flareup earlier this year as a result of Antibiotics . Because we understood eczema, we didn't fear the flareup. We knew exactly what the culprit was ( Antibiotics /candida & prescription antifungal cream) and responded accordingly. The proven cleansing and healing regimen remained the same. We immediately ceased the use of the antifungal cream, and we increased beta carotene and fiber foods to address the health of the immune system and to flush out the candida. The biggest problem with my nieces flareup was that it could have been avoided had my brother and sister-in-law heeded my warning not to use the prescription antifungal cream to combat the yeast infection developed as a result of the Antibiotic given to treat strep throat. I don't like " I told you so's" that involve my nieces suffering; as a result, I wasn't pleased that I was right regarding the medication. I didn't warn my brother and sister-in-law regarding the medication to boost my ego. The purpose of the warning was to avoid what I saw coming and didn't want my niece to suffer. Nonetheless, as quick as my brother and sister-in-law were to dismiss my warning; they were even quicker to relinquish my niece back to my care to remedy what had happened as a result of the total body flareup from the medication. Again, and due to the severity of her flareup, it took a little less than two weeks for her body to totally recover; yet, it recovered and healed just by feeding the immune system the nourishment needed to fight and by removing the allergen/irritant culprit. In essence, the most important thing to do to relieve my nieces new occurrence of an eczema flareup was to rid her body/environment of the thing promoting the flareup.
I believe, based upon your account in your posting, that your instincts regarding the type of eczema flaruep (contact dermatitis) and the cause of the flareup (carpet) is totally correct. In addition, John's account that an individual can experience several types of eczema in succession or simultaneously (with varying causes) is also correct. My point regarding the both of you is that you know what to do regarding the treatment of your children and their eczema flareups; as a result, don't discount your success based upon new flareups. It is most probably a new flareup based upon a new toxic contact.