the strong-field QED regime, in which the background strength is effectively of the order of $F_{cr}$, can be entered already at intensities of the order of $10^{23}\;\text{W/cm$^2$}$, if the laser field counterpropagates with respect to an electron/positron with energy of the order of $500\;\text{MeV}$.

In order to test QED in the strong-field regime by means of intense optical fields, it is essential that both experiments and theoretical predictions are correspondingly accurate. However, as it is understandable, first experiments in this regime have so far been designed especially to show the occurrence of phenomena like nonlinear Compton scattering [@Bula_1996], nonlinear Breit-Wheeler pair production [@Burke_1997; @Bamber_1999], and radiation reaction [@Cole_2018; @Poder_2018], without aiming at obtaining high-accuracy results. Correspondingly, on the theory side, the basic strong-field QED processes like nonlinear Compton scattering [@Goldman_1964; @Nikishov_1964; @Ritus_1985; @Baier_b_1998; @Ivanov_2004; @Boca_2009; @Harvey_2009; @Mackenroth_2010; @Boca_2011; @Mackenroth_2011; @Seipt_2011; @Seipt_2011b; @Dinu_2012; @Krajewska_2012; @Dinu_2013; @Seipt_2013; @Krajewska_2014; @Wistisen_2014; @Harvey_2015; @Seipt_2016; @Seipt_2016b; @Angioi_2016; @Harvey_2016b; @Angioi_2018; @Di_Piazza_2018_c; @Dinu_2018; @Alexandrov_2019; @Di_Piazza_2019; @Ilderton_2019_b] and nonlinear Breit-Wheeler pair production [@Reiss_1962; @Nikishov_1964; @Narozhny_2000; @Roshchupkin_2001; @Reiss_2009; @Heinzl_2010b; @Mueller_2011b; @Titov_2012; @Nousch_2012; @Krajewska_2013b; @Jansen_2013; @Augustin_2014; @Meuren_2015; @Meuren_2016; @Di_Piazza_2019; @King_2020] have been studied in detail at tree level by approximating the laser field as a plane wave (see also the reviews 