Using nonlinear simulations in two settings, we demonstrate that QG$^\mathrm{+1}$, a potential-vorticity based next-order-in-Rossby balanced model, captures several aspects of ocean submesoscale physics. In forced-dissipative 3D simulations under baroclinically unstable Eady-type background states, the statistical equilibrium turbulence exhibits long cyclonic tails and a plethora of rapidly-intensifying ageostrophic fronts. Despite that the model requires setting an explicit, small value for the fixed scaling Rossby number, the emergent flows are nevertheless characterized by vorticity and convergence values larger than the local Coriolis frequency, as observed in upper-ocean submesoscale flows. Simulations of QG$^\mathrm{+1}$ under the classic strain-induced frontogenesis set-up show realistic frontal asymmetry and a provable finite time blow-up, quantitatively comparable to simulations of the semigeostrophic equations. The inversions in the QG$^\mathrm{+1}$ model are straightforward linear Poisson problems, allowing for the reconstruction of all flow fields from the PV and surface buoyancy, while avoiding the semigeostrophic coordinate transformation. Taken together, these results suggest QG$^\mathrm{+1}$ might be a useful tool for studying upper-ocean submesoscale dynamics.