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Building upon recent results of Dubedat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations Ωδ to a simply connected domain Ω ⊂ ℂ we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on Ωδ as δ → 0. More precisely, let λ1,..., λn ∈ Ω and L be a macroscopic lamination on Ω \ {λ1,..., λn}, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities PLδ that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on Ωδ converge to a conformally invariant limit PL as δ → 0, for each L. Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety Hom(π1(Ω \ {λ1,..., λn}) → SL2(ℂ)) and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do not use any RSW-type arguments for double-dimers. The limits PL of the probabilities PLδ are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock-Goncharov lamination basis on the representation variety. The fact that PL coincides with the probability of obtaining L from a sample of the nested CLE(4) in Ω requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble. [ABSTRACT FROM AUTHOR]