Incremental text-to-speech (TTS) synthesis generates utterances in small linguistic units for the sake of real-time and low-latency applications. We previously proposed an incremental TTS method that leverages a large pre-trained language model to take unobserved future context into account without waiting for the subsequent segment. Although this method achieves comparable speech quality to that of a method that waits for the future context, it entails a huge amount of processing for sampling from the language model at each time step. In this paper, we propose an incremental TTS method that directly predicts the unobserved future context with a lightweight model, instead of sampling words from the large-scale language model. We perform knowledge distillation from a GPT2-based context prediction network into a simple recurrent model by minimizing a teacher-student loss defined between the context embedding vectors of those models. Experimental results show that the proposed method requires about ten times less inference time to achieve comparable synthetic speech quality to that of our previous method, and it can perform incremental synthesis much faster than the average speaking speed of human English speakers, demonstrating the availability of our method to real-time applications.