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Kidney exchange programs (KEPs) typically seek to match incompatible patient-donor pairs based on a utilitarian objective where the number or overall quality of transplants is maximized---implicitly penalizing certain classes of difficult to match (e.g., highly-sensitized) patients. Prioritizing the welfare of highly-sensitized (hard-to-match) patients has been studied as a natural \textit{fairness} criterion. We formulate the KEP problem as $k$-set packing with a probabilistic group fairness notion of proportionality fairness---namely, fair $k$-set packing (\f{}). In this work we propose algorithms that take arbitrary proportionality vectors (i.e., policy-informed demands of how to prioritize different groups) and return a probabilistically fair solution with provable guarantees. Our main contributions are randomized algorithms as well as hardness results for \f{} variants. Additionally, the tools we introduce serve to audit the price of fairness involved in prioritizing different groups in realistic KEPs and other $k$-set packing applications. We conclude with experiments on synthetic and realistic kidney exchange \textsc{FairSP} instances.