Systematic reviews synthesize biomedical evidence to inform clinical and public-health decision-making, but require domain experts to manually screen thousands of studies against detailed eligibility criteria. We investigate whether language models can assist with this bottleneck by learning from expert feedback on title and abstract screening decisions and model-generated reasoning. We introduce the Context-Aware Reasoning Framework (CARF), a human-in-the-loop framework that uses expert feedback on model-generated reasoning to adapt language models to context-dependent screening tasks. We evaluate CARF on systematic reviews of verbal autopsy and cause-of-death ascertainment methods used in global public health mortality surveillance. CARF shows competitive screening performance compared to evaluated models while retaining high sensitivity, and is more inclined to retain ambiguous, context-dependent studies for human review. Together, these findings suggest that expert-corrected reasoning is a promising supervision signal for context-dependent scientific screening, while retaining a role for human oversight.