This paper considers the achievable rate-exponent region of integrated sensing and communication systems in the presence of variable-length coding with feedback. This scheme is fundamentally different from earlier studies, as the coding methods that utilize feedback impose different constraints on the codewords. The focus herein is specifically on the Gaussian channel, where three achievable regions are analytically derived and numerically evaluated. In contrast to a setting without feedback, we show that a trade-off exists between the operations of sensing and communications.