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Detection and parameter inference of gravitational-wave signals \ncor{from compact mergers} rely on the comparison of the incoming detector strain data $d(t)$ to waveform templates for the gravitational-wave strain $h(t)$ that ultimately rely on the resolution of Einstein's equations via numerical relativity simulations. These, however, commonly output a quantity known as the Newman-Penrose scalar $ψ_4(t)$ which, under the Bondi gauge, is related to the gravitational-wave strain by $ψ_4(t)=\mathrm{d}^2h(t) / \mathrm{d}t^2$. Therefore, obtaining strain templates involves an integration process that introduces artefacts that need to be treated in a rather manual way. By taking second-order finite differences on the detector data and inferring the corresponding background noise distribution, we develop a framework to perform gravitational-wave data analysis directly using $ψ_4(t)$ templates. We first demonstrate this formalism, and the impact of integration artefacts in strain templates, through the recovery of numerically simulated signals from head-on collisions of Proca stars injected in Advanced LIGO noise. Next, we re-analyse the event GW190521 under the hypothesis of a Proca-star merger, obtaining results equivalent to those in Ref.[1], where we used the classical strain framework. We find, however, that integration errors would strongly impact our analysis if GW190521 was four times louder. Finally, we show that our framework fixes significant biases in the interpretation of the high-mass GW trigger S200114f arising from the usage of strain templates. We remove the need to obtain strain waveforms from numerical relativity simulations, avoiding the associated systematic errors.