We consider the Bessel-Hagen construction for linearised Weyl-squared gravity, whose action depends on the metric perturbation~$h_{\mu \nu }$ through the linearised Weyl tensor~$C_{\mu \nu \rho \sigma }^{(1)}$. This provides a curvature-based comparison with the Fierz-Pauli theory, in which the action contains first derivatives of~$h_{\mu \nu }$ whereas the first local gauge-invariant field strength contains two, and the construction determines a gauge-invariant class of Noether currents without selecting a preferred gauge-invariant local energy-momentum tensor. Since the Weyl tensor is the irreducible spin-2 part of the curvature, it is of particular interest to ask whether an action built from it admits a local representative analogous to the Maxwell tensor. We show that the Bessel-Hagen construction extends to the conformal group, acting on the fields on a fixed Minkowski background, and yields a gauge-invariant conserved-current class for each generator. Nonetheless, no nonzero local, polynomial, symmetric rank-two tensor of mass dimension four, quadratic in $h_{\mu \nu }$ and with constant dimensionless coefficients and no explicit coordinate dependence, is both strictly invariant under the linearised diffeomorphism and Weyl gauge symmetries and conserved on the Bach shell. The gauge symmetries restrict such a tensor to depend on~$C^{(1)}$ and its derivatives, while the dimension and field degree permit only terms quadratic in~$C^{(1)}$ without further derivatives. Four-dimensional Weyl identities then reduce all candidates to pure traces, whose conservation requires their coefficients to vanish. Thus, building the action from a gauge-invariant curvature does not recover the electromagnetic result. A first-order gauge-covariant field strength may provide a route to such a representative, although its presence alone is not sufficient.