Mixed-criticality applications, such as satellite terminals, industrial telemetry, embedded systems, tactical, and other constrained links, often multiplex a small, latency-critical message class and bulk traffic over a single commodity transport connection. A single FIFO connection can starve the critical class under load. The obvious alternative, opening parallel connections, costs an additional five-tuple (often blocked by carrier-grade NAT, port budgets, and operator policy) and is not always available; when the critical class is light, two connections can also be bandwidth-fair only in aggregate rather than single-flow fair. We present CATS (Conductor-driven Asymmetric Transport Scheme), a sender-side, receiver-transparent transport-layer priority scheme over TCP: a Conductor assigns each message a priority class and just-in-time sequence numbers, using a credit-based shaper. CATS provides the one combination its alternatives cannot: deterministic non-starvation together with single-flow fairness, plus a provable bounded per-class delay. We then show that, crucially, CATS-over-TCP is not a tail-latency mechanism, and why. Three structural barriers bound in-band priority: the in-order sequence space (head-of-line blocking), the shared congestion window (cross-class coupling), and the per-flow granularity of network QoS (in-band priority is invisible to it). These barriers explain why fair-queuing and even the modern low-latency standard L4S cannot help a single connection, and why two parallel connections reduce the latency tail at the cost of an additional flow. We give CATS-over-QUIC as the principled escape: independent streams with per-stream isolation under aggregate-coupled congestion control self-isolate at the endpoint, attaining the guarantees on one fair flow. An ns-3 evaluation and QUIC proof-of-concept support the findings.