Modern Artificial Intelligence (AI) workloads deployed across the heterogeneous tiers of an edge--cloud continuum must satisfy multi-dimensional Service Level Objectives (SLOs) over latency, throughput, and output quality. For each incoming task, the scheduler picks both a target node and a processing mode (e.g., full or reduced inference precision). We call this class of problems \emph{Continuous Multi-Mode Scheduling} (CMMS). Comparing CMMS algorithms fairly is difficult because prior studies typically evaluate each controller in its own stack, under a single workload, and without reporting per-decision overhead. To close these gaps, we present an open source benchmark platform that features (i) a unified controller interface, (ii) a closed-loop workload driver covering multiple workload patterns, and (iii) dual-metric SLO scoring that reports raw SLO (overall compliance) and steady-state SLO (compliance during stable operation) separately. Running six controllers across five cluster configurations and two load regimes (424 episodes), we find that controller rankings are strongly configuration-dependent: a deep reinforcement-learning winner under light workloads loses to a rule-based heuristic by nearly 29 percentage points once load intensifies, at roughly 500$\times$ the per-decision operational overhead. We further show that separating raw from steady-state SLOs exposes switching costs that a single aggregate score would otherwise conflate.