Efficient GPU execution of convolution operators is governed by memory-access efficiency, on-chip data reuse, and execution mapping rather than arithmetic throughput alone. This paper presents a controlled operator-level study of CUDA kernel optimization for the depthwise convolution used in Structured State Space Model Convolutional Diagonal (S4ConvD), together with a cloud-compatible, counter-free performance analysis methodology. The operator, model, dataset, and training configuration are fixed, and only the CUDA kernel implementation is varied. The evaluated CUDA kernels comprise naive, global-memory-coalesced, shared-memory cache-blocked, and warp-tiled variants, covering forward, input-gradient, and weight-gradient execution paths under steady-state training conditions. Performance is characterized using a counter-free methodology that combines CUDA-event timing, execution-path decomposition, analytically derived memory-traffic modeling, effective-bandwidth estimation, and roofline analysis. This enables profiling-like architectural insights without requiring hardware performance counters or privileged profiling access. The warp-tiled kernel reduces convolution runtime by $3.26\times$ relative to the naive CUDA baseline, while end-to-end training speedup reaches $1.29\times$. A PyTorch implementation is used separately for numerical validation and runtime context, but is not treated as a controlled architectural baseline. Forward and input-gradient paths benefit substantially from improved locality and on-chip data reuse, whereas the reduction-dominated weight-gradient path remains the primary bottleneck. The results demonstrate that meaningful architecture-level GPU kernel analysis can be performed reproducibly in restricted cloud environments, even without access to hardware performance counters.