Fe-based superconductors and in particular K-doped BaFe2As2 (K-Ba122) are materials of interest for possible future high-field applications. However the critical current density (Jc) in polycrystalline Ba122 is still quite low and connectivity issues are suspected to be responsible. In this work we investigated the properties of high-purity, carefully processed, K-Ba122 samples synthesized with two separate heat treatments at various temperatures between 600 and 825 °C. We performed specific heat characterization and Tc-distribution analysis up to 16 T and we compared them with magnetic Tc and Jc characterizations, and transmission-electron-microscopy (TEM) microstructures. We found no direct correlation between the magnetic Tc and Jc, whereas the specific heat Tc-distributions did provide valuable insights. In fact the best Jc-performing sample, heat treated first at 750 °C and then at 600 °C, has the peak of the Tc-distributions at the highest temperatures and the least field sensitivity, thus maximizing Hc2. We also observed that the magnetic Tc onset was always significantly lower than the specific heat Tc: although we partially ascribe the lower magnetization Tc to the small grain size (< λ, the penetration depth) of the K-Ba122 phase, this behaviour also implies the presence of some grain-boundary barriers to current flow. Comparing the Tc-distribution with Jc, our systematic synthesis study reveals that increasing the first heat treatment above 750 °C or the second one above 600 °C significantly compromises the connectivity and suppresses the vortex pinning properties. We conclude that high-purity precursors and clean processing are not yet enough to overcome all Jc limitations. However, our study suggests that a higher temperature Tc-distribution, a larger Hc2 and a better connectivity could be achieved by lowering the second heat treatment temperature below 600 °C thus enhancing, as a consequence, Jc.