End-to-end automatic speech recognition (E2E-ASR) has the potential to improve performance, but a specific issue that needs to be addressed is the difficulty it has in handling enharmonic words: named entities (NEs) with the same pronunciation and part of speech that are spelled differently. This often occurs with Japanese personal names that have the same pronunciation but different Kanji characters. Since such NE words tend to be important keywords, ASR easily loses user trust if it misrecognizes them. To solve these problems, this paper proposes a novel retraining-free customized method for E2E-ASRs based on a named-entity-aware E2E-ASR model and phoneme similarity estimation. Experimental results show that the proposed method improves the target NE character error rate by 35.7% on average relative to the conventional E2E-ASR model when selecting personal names as a target NE.