Tensor parallelism (TP) in large-scale LLM inference and training introduces frequent collective operations that dominate inter-GPU communication. While in-switch computing, exemplified by NVLink SHARP (NVLS), accelerates collective operations by reducing redundant data transfer, its communication-centric design philosophy introduces the mismatch between its communication mode and the memory semantic requirement of LLM's computation kernel. Such a mismatch isolates the compute and communication phases, resulting in underutilized resources and limited overlap in multi-GPU systems. To address the limitation, we propose CAIS, the first Compute-Aware In-Switch computing framework that aligns communication modes with computation's memory semantics requirement. CAIS consists of three integral techniques: (1) compute-aware ISA and microarchitecture extension to enable compute-aware in-switch computing. (2) merging-aware TB (Thread Block) coordination to improve the temporal alignment for efficient request merging. (3) graph-level dataflow optimizer to achieve a tight cross-kernel overlap. Evaluations on LLM workloads show that CAIS achieves 1.38$\times$ average end-to-end training speedup over the SOTA NVLS-enabled solution, and 1.61$\times$ over T3, the SOTA compute-communicate overlap solutions but do not leverage NVLS, demonstrating its effectiveness in accelerating TP on multi-GPU systems.