Fine-tuning generic ASR models with large-scale synthetic personal data can enhance the personalization of ASR models, but it introduces challenges in adapting to synthetic personal data without forgetting real knowledge, and in adapting to personal data without forgetting generic knowledge. Considering that the functionally invariant path (FIP) framework enables model adaptation while preserving prior knowledge, in this letter, we introduce FIP into synthetic-data-augmented personalized ASR models. However, the model still struggles to balance the learning of synthetic, personalized, and generic knowledge when applying FIP to train the model on all three types of data simultaneously. To decouple this learning process and further address the above two challenges, we integrate a gated parameter-isolation strategy into FIP and propose a knowledge-decoupled functionally invariant path (KDFIP) framework, which stores generic and personalized knowledge in separate modules and applies FIP to them sequentially. Specifically, KDFIP adapts the personalized module to synthetic and real personal data and the generic module to generic data. Both modules are updated along personalization-invariant paths, and their outputs are dynamically fused through a gating mechanism. With augmented synthetic data, KDFIP achieves a 29.38% relative character error rate reduction on target speakers and maintains comparable generalization performance to the unadapted ASR baseline.