KLM & BA both stop in Saudi between Qatar & Europe, and there's never a problem with visa, as there shouldn't be if you're not going through their immigration. The airline should be able to tell you, as they have a legal responsibility to avoid carrying unwelcome aliens.
A problem you perhaps haven't thought of is booze. I've heard stories of transit passengers'duty-frees being smashed by abusive Saudi matawas with nothing better to do.