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Multi-site clinical initiatives within infrastructures like the Swiss Personalized Health Network (SPHN) are essential for impactful research, but privacy regulations and significant inter-institutional heterogeneity pose major barriers to data sharing. The SPHN-LUCID NDS (National Data Stream) project, which creates a FAIR-principles-based cohort to monitor low-value care for medical inpatients across five Swiss university hospitals, exemplifies this real-world challenge. Our study investigates the viability of using Differentially Private (DP) synthetic data generation to create a high-fidelity, shareable version of this sensitive and complex dataset. We generated multiple synthetic cohorts using a range of privacy-loss budgets, allowing for a direct analysis of the trade-off between privacy guarantees and data utility. We evaluated the preservation of local, per-hospital heterogeneous data patterns, which is the key challenge for any federated analysis. This work provides a practical measure of the specific impact of the privacy budget on the statistical fidelity of a real-world, multi-site clinical dataset.