Serverless platforms face a trade-off: conventional cluster managers like Kubernetes offer compatibility for co-locating Function-as-a-Service (FaaS) and Backend-as-a-Service (BaaS) components of serverless applications, at the cost of high cold-start latency, whereas specialized FaaS-only systems like Dirigent achieve low latency by sacrificing compatibility, preventing integrated management and optimization. Our analysis reveals that FaaS traffic is bimodal: predictable, sustainable traffic consumes >98% of cluster resources, whereas sporadic, excessive bursts stress the control plane's scaling latency, not its throughput. With these insights, we design PulseNet, a serverless architecture that uses a dual-track control plane tailored to both traffic types. PulseNet's standard track manages sustainable traffic with long-lived, full-featured Regular Instances under a conventional cluster manager, preserving compatibility for the majority of the workload. To handle excessive traffic, an expedited track bypasses the slow manager to rapidly create short-lived, disposable Emergency Instances, minimizing cold-start latency and resource waste from idle instances. This hybrid approach achieves 35% better performance than Dirigent, a FaaS-only system, on a production workload at the same cost and outperforms other Kubernetes-compatible systems by 1.5-3.5x, reducing the cost by up to 70%.