Carpets can be cleaned however there is still dirt, dust, and fibers in the air. I have seen what is under them when carpets are removed. Also cleaning the air ducts is a good idea. Also, maybe look under all sinks and any area with a water source for possible leaks that are growing mildew. That is all I have…

Our flooring company charges a little more but will do an occupied unit and move the furniture, tenants just have to empty/ box personal belongings... good luck sounds like you're doing everything you can

I rented a house and had my carpets steam cleaned. The bedroom my office is in was horrible. I would have headaches & be congested the minute I walked in. I hired another carpet company to come out (was only this room that bothered me) and they figured out the previous company had used too much solution & too much water (there was a sickly sweet smell in the room). I had them shampoo again and it hasn't bothered me since.

We had a similar situation and ended up cleaning the ducts. It definitely helped. That being said, I've also seen this issue in a unit with mold that wasn't visible.

I had a person I worked with who was allergic to the carpenters glue used to assemble her cabinets in her new house.
Big $$$ to find that issue.
Offer transfer or if none available consider a lease break at no cost, just 30 days notice. It could work out to be the least expensive option.

You mentioned it’s his second lease term - is that in the same unit? If so… was this unit closed up for a period of time while he was home?
If it was… was the AC kept at a certain temp? (I’m in Florida asking this, fyi) Could there be “discoloration” somewhere that hasn’t been noticed? Was it a little dusty?