Consider calling a carpet cleaning service that also cleans duct work (everything coming from your furnace, vents, etc). It's amazing what builds up in ducts and vents.
Also, contact an exterminator who can thoroughly and professionally determine if there are fleas or invisible things that your daughter might be reacting to. Most homeowners can't detect the bugs that a pro can.
These "bites" may not be bites at all - they could be an allergic reaction to nearly anything. Start keeping track of where she plays, where she is during the day when the welts appear, and most importantly, what she eats and drinks. Be specific.
Your vacuuming and shampooing the carpets could be a factor - find out if your vacuum has a filter that traps even the tiniest pollutants, and find out what is in the carpet shampoo.
A dust mite allergy is a real thing, but since she rarely gets bites at night, and dust mites love bedding, that doesn't sound plausible. Try switching out all cleaners and cleansers to unscented products (and don't get fragrance-free; they're not unscented) that are all natural and safe for sensitive skin. Try all cotton clothing.
And definitely get her to a dermatologist and an allergist for testing and evaluation.