The higher fusion level logarithmic minimal models LM(P; P'; n) have recently been constructed as the diagonal GKO cosets (A(1) 1)k (A(1) 1)n/(A (1) 1)k+n where n ≥ 1 is an integer fusion level and k = nP/(P' - P) - 2 is a fractional level. For n = 1, these are the well-studied logarithmic minimal models LM(P, P') ≡ LM(P; P'; 1). For n = 2, we argue that these critical theories are realized on the lattice by n × n fusion of the n = 1 models. We study the critical fused lattice models LM(p, p')n×n within a lattice approach and focus our study on the n = 2 models. We call these logarithmic superconformal minimal models LSM(p, p') ≡ LM(P, P'; 2) where P = |2p - p'|, P' = p' and p' p' are coprime. These models share the central charges c = cP;P';2 = 3/2 (1 - 2(P' - P)2=PP') of the rational superconformal minimal models SM(P, P'). Lattice realizations of these theories are constructed by fusing 2 × 2 blocks of the elementary face operators of the n = 1 logarithmic minimal models LM(p, p'). Algebraically, this entails the fused planar Temperley-Lieb algebra which is a spin-1 Birman-Murakami-Wenzl tangle algebra with loop fugacity β2 = [x]3 = x2 +1+x-2 and twist ω = x4 where x = e iλ and λ = (p' - p)π/p'. The first two members of this n = 2 series are superconformal dense polymers LSM(2, 3) with c = -5/2, β2 = 0 and superconformal percolation LSM(3, 4) with c = 0, β2 = 1. We calculate the bulk and boundary free energies analytically. By numerically studying finite-size conformal spectra on the strip with appropriate boundary conditions, we argue that, in the continuum scaling limit, these lattice models are associated with the logarithmic superconformal models LM(P, P'; 2). For system size N, we propose finitized Kac character formulae of the form q -cPP';2/24+δP,P';2 r;s;ℓ Χ̃ (N) r;s;ℓ(q) for s-type boundary conditions with r = 1, s = 1, 2, 3;⋯, ℓ = 0, 1, 2. The P; P' dependence enters only in the fractional power of q in the prefactor and ℓ = 0, 2 label the Neveu-Schwarz sectors (r + s even) and ℓ = 1 labels the Ramond sectors (r + s odd). Combinatorially, the finitized characters involve Motzkin and Riordan polynomials defined in terms of q-trinomial coeficients. Using the Hamiltonian limit and the finitized characters, we argue, from examples of finite-lattice calculations, that there exist reducible yet indecomposable representations for which the Virasoro dilatation operator L0 exhibits rank-2 Jordan cells, confirming that these theories are indeed logarithmic. We relate these results to the N = 1 superconformal representation theory.