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 potential energy surfaces, the role of the heat bath, in particular, in a realistic form of the system-bath interaction has not been well explored. Previous studies are largely based on a one-dimensional model and linear-linear 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-RPV 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 the hierarchical equations of motion approach. We analyze the role of the heat bath interaction in the chemical reaction rate as a function of the system-bath coupling strength at different temperatures 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. The 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.