Are there specific conventional DMARDs and/or biologics that are contraindicated in patients with alpha-gal allergy?  

To my knowledge and after review of the literature, no.

Here is a Q&A from AAAAI (American Academy of Allergy, Asthma and Immunology) addressing this question for etanercept and other monoclonal antibodies that are manufactured in cell lines that glycosylate with alpha-gal. The author concludes, "The risk is not zero, but likely very low".