Large language models (LLMs) have transformed machine translation, yet they have a high subword fertility issue for low-resource languages, which leads to slow inference speed and increased costs. While vocabulary expansion via continual pre-training is a common solution, it often degrades translation quality and requires large target-language corpora, which are unavailable for truly low-resource languages. To address this, we investigate tokenization efficiency through an information-theoretic lens, building on the established hypothesis that word length correlates with information content. From this perspective, we characterize tokenization inefficiency as having high fertility for low-information (highly predictable) words. Guided by this principle, we introduce a novel fine-tuning strategy that systematically identifies informationally redundant words—those with high fertility but low information content—for targeted vocabulary expansion and model fine-tuning. Experiments fine-tuning BLOOM and LLaMA-3 in English-Manipuri and other two language pairs show that our proposed method significantly reduces fertility by 50\% and accelerates inference by more than 2 times, without compromising and often exceeding the translation quality of standard LLM baselines, providing a theoretically grounded solution for efficient LLM-based MT.