Accelerator-based heterogeneous architectures have become the cornerstone of modern artificial intelligence (AI) applications, spanning from embedded devices to high-performance desktop and server systems. On such platforms, hard real-time (HRT) and soft real-time (SRT) tasks often coexist, where HRT tasks require strict deadline guarantees while SRT tasks benefit from adaptive scheduling to improve responsiveness. However, runtime priority tuning on heterogeneous systems is inherently unsafe, as even minor changes may invalidate HRT schedulability under complex inter-segment interference and accelerator-side contention. This paper presents MiRTS, an offline-online co-designed scheduling framework that enables safe online tuning to improve SRT performance and also provides provable HRT guarantees on heterogeneous platforms. MiRTS first performs offline analysis to identify HRT-feasible priority configurations, and then encodes the certified safe tuning space into a Guaranteed Priority Tree (GPT). Rather than performing costly on-the-fly schedulability analysis, MiRTS restricts runtime adjustments to verified transitions within GPT and guides them using a hybrid reactive-proactive strategy based on runtime measurements. In this way, MiRTS improves SRT timing performance while preserving HRT schedulability by construction. Experimental results demonstrate that MiRTS significantly outperforms state-of-the-art methods, achieving up to 75.0% improvement in HRT acceptance ratio and a 65.3% reduction in SRT relative response time.