There is a field called $C^{0}$-symplectic topology, which studies properties stable with respect to convergence of Lagrangians under $C^{0}$-Hamiltonian isotopies, especially spectral type invariants. This is morally related to our concerns here, but as far as the author understands, Floer theory for Lagrangian varifolds/currents is not yet explicitly treated in this field.