What biologics would you use in a patient with rheumatoid arthritis who developed multiple basal cell carcinomas requiring Mohs surgery while on a TNF-inhibitor and did not respond to rituximab?

It would be a discussion with the patient about risks and benefits if the RA is well controlled on a TNF-inhibitor. If the patient wanted to switch, I'd consider switching to an IL-6R inhibitor like tocilizumab or sarilumab.

Why is it an issue? TNF inhibitors do not promote BCCs.

If you have data that they do, cite the study, please.