The continuous-time joint replenishment problem has long served as a foundational inventory management model. Even though its unconstrained setting has seen recent algorithmic advances, the incorporation of resource constraints into this domain precludes the application of newly discovered synchronization techniques. Such constraints arise in a broad spectrum of practical environments where resource consumption is bounded as an aggregate rate over time. However, for nearly four decades, the prevailing approximation guarantee for resource-constrained joint replenishment has remained $\frac{ 1 }{ \ln 2 } \approx 1.4427$, achieved via classical power-of-$2$ policies. In this paper, we circumvent these structural policy restrictions by devising generalized rounding frameworks, demonstrating that a well-known convex relaxation is much tighter than previously established. In particular, we expand our analytical scope to encompass fractional base expansion factors, randomized shifting, and staggered interleaved grids. Through this multifaceted methodology, we present a sequence of gradually improving performance guarantees. First, by proposing a best-of-two framework that exploits structural asymmetries between deterministic power-of-$m^{1/k}$ policies, we surpass the classical barrier to obtain a $1.3776$-approximation. Second, by injecting a random shift into the logarithmic grid domain and formulating a factor-revealing linear program to optimize a dual-policy approach, we attain a $1.2512$-approximation. Finally, by superimposing a secondary offset grid to subdivide rounding intervals and suppress holding cost inflation, we utilize interleaved policies to arrive at our ultimate approximation ratio of $\frac{5}{6\ln 2} \approx 1.2023$, which is proven to be best-possible for the class of interleaved power-of-$m^{1/k}$ policies.