Recent research suggests policy instrument design and communicated effectiveness influence both perceived effectiveness and acceptance. However, directions of causality remain debated. Traditional models posit that perceiving a policy as effective increases acceptance—alongside perceptions of personal benefit and fairness. Contrastingly, alternative accounts argue that perceived effectiveness rather follows a supportive or opposing attitude - in turn influenced by more salient policy perceptions, namely personal and distributional consequences - rather than the other way around. Additionally, general climate policy support (GCPS) may moderate how design features affect acceptance, but this remains understudied. This study investigates (1) whether these competing causal pathways are at play, and (2) the moderating role of GCPS. We conducted a factorial survey experiment on heating sector policy instruments (N=2,579), varying instrument design and effectiveness information. Using random effects regression and (multi-group) structural equation modeling, we find evidence for both pathways—conditional on GCPS. Only among general climate policy supporters, optimistic effectiveness information increased acceptance via perceived effectiveness. However, for both supporters and opponents, policies designed to have positive personal and fair distributional consequences were also seen as more effective—mediated by acceptance. These findings suggest bidirectional dynamics between perceived effectiveness and acceptance and thus question dominant assumptions in climate policy literature. For policymakers, these findings imply that building broad support requires fairness and limited personal costs first, as these factors not only drive acceptance but also contribute to whether policy instruments are judged effective.