The Hybrid Total Finite Element Tearing and Interconnecting (HTFETI) method plays an important role in solving large-scale and complex engineering problems. This method needs to handle numerous matrix-vector multiplications. Directly calling the vendor-optimized library for general matrix-vector multiplication (gemv) on GPU leads to low performance, since it does not consider optimizations for different matrix sizes in HTFETI, i.e. different row and column sizes. In addition, state-of-the-art graph partitioning methods cannot guarantee load balancing for HTFETI, since the matrix size is determined by the length of the subdomain boundary. To solve the problems above, we first port gemv to the multi-stream pipeline scheme and develop a new batched kernel function on GPU, which brings 15%~30% throughput improvement and 37% average GFLOPs improvement, respectively. We also propose a multi-grained load-balancing scheme based on graph repartitioning and work-stealing, and the load imbalance ratio is down to 1.05$\sim$1.09 from 1.5. We have successfully applied the scalable HTFETI method to simulate the whole core assembly of China Experimental Fast Reactor (CEFR) for steady-state analysis, and the efficiencies of weak scalability and strong scalability reach 78% and 72% on 12,288 GPUs, respectively. As far as we know, this is the first time that HTFETI has been used in large-scale and high-fidelity whole core assembly simulation.