Battery-powered edge systems must operate under tight energy budgets while facing growing computational demand from rapidly evolving edge workloads. Field-programmable gate arrays (FPGAs) offer middle ground when optimized for energy, especially flash-based FPGAs due to inherent low-power characteristics. Microchip flash-based SoC FPGAs further expose ultra-low-power (LP) modes including fabric Flash*Freeze (F*F), processor sleep and selectable standby clocks. Combining these modes with HW/SW partitioning and clock-frequency scaling can reduce energy for low-duty-cycle workloads; however, selecting an energy-efficient operating point in this multidimensional design space is non-trivial. This work explores the design space by measuring and analyzing LP modes across three architectural approaches (SW, co-design, and HW) under frequency scaling on a Microchip Smartfusion2 platform, using a low-duty-cycle heart-rate monitoring workload. Measurements indicate that, for low-duty-cycle workloads, total energy is dominated by the idle phase and is minimized by combining fabric-F*F with processor sleep. The results further show that main-clock downscaling reduces active-phase current but has limited impact on idle consumption once F*F and sleep are applied, while standby-clock selection trades idle current against LP entry/exit latency. Event-rate scaling further shows that the energy-optimal operating point can shift with duty cycle. We provide measurement-based guidelines for duty-cycle-aware energy-efficient operating point selection in similar flash-based SoC platforms.