Deploying tiny object perception on edge platforms is challenging because practical systems must satisfy both strict compute budgets and end-to-end latency constraints. A common strategy is to first select a small number of candidate patches from a high-resolution image and then apply downstream processing only to the selected regions. However, existing detector-based frontends are not well aligned with this setting: strong offline detection accuracy does not necessarily yield effective low-budget patch prioritization, nor does it guarantee usable performance once transport and inference delays are considered. In this work, we study budgeted tiny object selection on edge platforms from a joint algorithm--system perspective. We present DenseScout, a lightweight dense-response selector with only 1.01M parameters, which directly ranks candidate patch locations from a high-resolution scene via a lightweight proxy input and is better aligned with low-budget tiny-object prioritization than detector-style frontends. To bridge offline selector quality and deployable utility, we further develop a transport-aware runtime realization on heterogeneous edge devices and adopt QoS-constrained recall, which counts a target as successfully perceived only if it is covered by the selected regions and the end-to-end processing finishes before the deadline. Experiments show that DenseScout consistently outperforms detector-based baselines in offline budgeted patch-selection evaluation, especially in low-budget regimes, while cross-platform results on RK3588 and Jetson Orin NX show that deployable performance depends jointly on selector quality and runtime realization efficiency. These results suggest that edge tiny object perception should be optimized as an algorithm--system co-design problem rather than as isolated model selection.