Advances in computer science, including machine learning and automatic differentiation (AD), offer the opportunity to revisit classic turbulence models equipped with a new set of optimization tools. In particular, gradient descent-based optimization techniques used together with differentiable solvers allow to iteratively tune the coefficients of a model based on the solver's output and its derivative with respect to input parameters, a.k.a. solver-in-the-loop approach [1, 2]. In the context of large-eddy simulation (LES), a subgrid-scale stress model can be directly trained together with the solver, hence avoiding a-priori assumptions of the filter length scale that can yield poor results during inference (a-posteriori evaluation). Instead of explicitly modelling the subgrid-scale stresses, we exploit AD to tune the coefficients of a numerical scheme which ultimately acts as a LES filter, a.k.a. implicit LES modelling. This framework benefits from avoiding the definition of an arbitrary LES filter length scale, can work for any parametrized numerical discretization scheme, and yields optimal SGS dissipation. The one-dimensional viscous Burgers' equation is used as a simple test case of nonlinear multiscale convection-diffusion phenomena. We use a finite-volume approach with flux reconstruction defined by the $\kappa$-schemes family by van Leer [3]. The optimization problem focuses on recovering the analytical energy decay rate in wavenumber $(k)$ space, $E\propto k^{-2}$, by tuning the $\kappa$ parameter based on AD. Importantly, this stabilises the solution by removing Gibbs-like oscillations near the grid cut-off wavenumber at the expense of added numerical dissipation. In contrast, smooth solutions arising from low-Reynolds-number cases yield positive $\kappa$ values, thus recovering high-order reconstruction schemes. The combined effect of $\kappa$ with flux limiters is also explored. It is found that when stabilization is provided by the limiter, the optimization problem correctly finds high values of $\kappa$. This allows extending the inertial subrange by removing the right amount of energy piling up near the grid cut-off wavenumber.