We continue the effort of defining and evaluating the quantum entropy function for supersymmetric black holes in 4d ${cal N} = 2$ gauged supergravity, initiated in [1803.05920]. The emphasis here is on the missing steps in the previous localization analysis, mainly dealing with one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact space $mathbb{H}_2 times Sigma_{rm g}$ with particular boundary conditions. We use several different techniques to arrive at consistent results, which have a most direct bearing on the logarithmic correction terms to the Bekenstein-Hawking entropy of said black holes.
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