L.R.
If both hands are swollen -- and not just one hand -- I would question whether this is from bug bites at all. Do you actually see bite marks? Small, red, raised, obvious location of the original bite? If not, I would see a different pediatrician (it sounds like yours might have seen one bite affect her once and is assuming all other subsequent symptoms are also bites, which is not good doctoring) and I also would tell the doctor -- not ask, tell -- that you want a referral to a pediatric allergist for tests.
If you are not seeing any other evidence of spiders or mosquitos in your home, really, I think it's past time to get some other opinions. Could be a contact allergy to something she plays with. And swelling can be a symptom of many and very different conditions that aren't allergies, too, so go beyond the pediatrician to get it checked out. I hope she gets better soon.