Hello. Thank you for your query at DoctorSpring.com.
It is highly unlikely. On the other hand, intercourse during menstruation puts the woman at a higher risk of acquiring infections. The risk of acquiring sexually transmitted diseases and infections is higher than normal during this time because the cervix opens to allow blood to pass through. Unfortunately, this creates the perfect pathway for bacteria to travel deep inside your pelvic cavity. She is also more likely to develop yeast or bacterial infections because the vagina's pH during menstruation is less acidic.
So safe and hygienic practices are even more important here.
At the same time, the man too has an increased risk since the woman is also more likely to pass on blood-borne diseases like HIV and Hepatitis, if she is positive, to a partner during her period. However, intercourse during menstruation causing a kidney infection is unlikely.
Since you are both negative for STDs, transmission of these is also ruled out.
As for the right lower back pain, that too is unlikely to be from a kidney infection. It's more likely to be due to muscle strain. Infection in the kidney presents with fever with chills, nausea, vomiting, severe lower back pain and other symptoms of urinary tract infection like burning/pain while passing urine, increased frequency etc.
If your husband does not have any of the other symptoms, then a kidney infection is not at all likely. Please assure him of the same. If he isn't convinced, a simple urine analysis test can be done. A normal result will rule out any infection.
I hope this was helpful. Feel free to ask queries.
Regards.