Getting Rid of Unseen Biting Bugs?

We live in an older town home with what is probably the original carpeting. Our 5 year old keeps getting bit at night by a mystery bug. They look like mosquito bites (round, raised areas that itch, but have no distinguishable bite mark), but they're not mosquito bites. So far, we've ruled out bed bugs, but at this point, I'll try anything.

So far, we've treated the baseboards for bedbugs and carpet beetles, treated his entire carpet with diatomaceous earth, put citronella around the window and under his bed, started using a mattress cover, and still he is getting bitten. These bites are a problem. Some nights, it's all over his back, other nights he'll get them on his forehead.

We're not wealthy people and because this isn't our house, we can't rip up or replace the carpeting. What else can I try? We don't have the spare funds to fumigate our entire upstairs and I feel like a failure for letting something so small bite my kid. Help!