Directional quantum amplifiers enable weak-signal readout while protecting the source, but reservoir noise limits sensitivity. Finite-band suppression must reconcile frequency-dependent responses with fixed local couplings and finite rates. We construct adjacent-site loss channels attaining the known single-frequency minimum under local compatibility conditions, as an uncapped benchmark. For specified uniform and dimerized chains, we compare fixed reservoirs at matched drift, ports, high-gain band and complete gain/loss rate caps. At the representative cap R=8 and halfwidth Ω=0.05, nearest-neighbor arithmetic band average noise exceeds the unrestricted optimum by more than 1.01265 quanta in the uniform chain for every finite N≥64. Boundary-response bounds give necessary rates and targets requiring range two. At N=64, finite-auxiliary calculations retain a noise advantage under matched total coupling budgets; response bounds quantify noise-resolution tolerances. These theoretical results guide reservoir design and noise calibration for a proposed cryogenic microwave array with a 100 kHz design band.