Recent studies have extensively explored NPU architectures for accelerating AI inference in on-device environments, which are inherently resource-constrained. Meanwhile, transformer-based large language models (LLMs) have become dominant, with rapidly increasing model sizes but low degree of parameter reuse compared to conventional CNNs, making end-to-end execution on resource-limited devices extremely challenging. To address these challenges, we propose TriGen, a novel NPU architecture tailored for resource-constrained environments through software-hardware co-design. Firstly, TriGen adopts low-precision computation using microscaling (MX) to enable additional optimization opportunities while preserving accuracy, and resolves the issues that arise by employing such precision. Secondly, to jointly optimize both nonlinear and linear operations, TriGen eliminates the need for specialized hardware for essential nonlinear operations by using fast and accurate LUT, thereby maximizing performance gains and reducing hardware-cost in on-device environments, and finally, by taking practical hardware constraints into account, further employs scheduling techniques to maximize computational utilization even under limited on-chip memory capacity. We evaluate the performance of TriGen on various LLMs and show that TriGen achieves an average 2.73x performance speedup and 52% less memory transfer over the baseline NPU design with negligible accuracy loss.