Given the crucial role of climate in malaria transmission, many mechanistic models of malaria represent vector biology and the parasite lifecycle as functions of climate variables in order to accurately capture malaria transmission dynamics. Lower dimension mechanistic models that utilize implicit vector dynamics have relied on indirect climate modulation of transmission processes, which compromises investigation of the ecological role played by climate in malaria transmission. In this study, we develop an implicit process-based malaria model with direct climate-mediated modulation of transmission pressure borne through the Entomological Inoculation Rate (EIR). The EIR, a measure of the number of infectious bites per person per unit time, includes the effects of vector dynamics, resulting from mosquito development, survivorship, feeding activity and parasite development, all of which are moderated by climate. We combine this EIR-model framework, which is driven by rainfall and temperature, with Bayesian inference methods, and evaluate the model's ability to simulate local transmission across 42 regions in Rwanda over four years. Our findings indicate that the biologically-motivated, EIR-model framework is capable of accurately simulating seasonal malaria dynamics and capturing of some of the inter-annual variation in malaria incidence. However, the model unsurprisingly failed to reproduce large declines in malaria transmission during 2018 and 2019 due to elevated anti-malaria measures, which were not accounted for in the model structure. The climate-driven transmission model also captured regional variation in malaria incidence across Rwanda's diverse climate, while identifying key entomological and epidemiological parameters important to seasonal malaria dynamics. In general, this new model construct advances the capabilities of implicitly-forced lower dimension dynamical malaria models by leveraging climate drivers of malaria ecology and transmission.