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We consider a proton-transfer (PT) system described by a proton-transfer reaction (PTR) coordinate and a rate promoting vibrational (RPV) coordinate interacting with a non-Markovian heat-bath. While dynamics of PT processes has been widely discussed using two-dimensional (2D) potential energy surfaces (PES), the role of the heat-bath, in particular, for a realistic form of the system-bath interaction has not been well explored. Previous studies are largely based on one-dimensional model and linear-linear (LL) system-bath interaction. In the present study, we introduce an exponential-linear (EL) system-bath interaction, which is derived from the analysis of a PTR-PRV system in a realistic situation. This interaction mainly causes vibrational dephasing in the PTR mode and population relaxation in the RPV mode. Numerical simulations were carried out using hierarchy equations of motion approach. We analyze the role of the heat-bath interaction on the chemical reaction rate as a function of the system-bath coupling strength at different temperature and for different values of the bath correlation time. A prominent feature of the present result is that while the reaction rate predicted from classical and quantum Kramers theory increases as the temperature increases, the present EL interaction model exhibits opposite temperature dependence. Kramers turn-over profile of the reaction rate as a function of the system-bath coupling is also suppressed in the present EL model turning into a plateau-like curve for larger system-bath interaction strength. Such features arise from the interplay of the vibrational dephasing process in the PTR mode and the population relaxation process in the RPV mode.