We show how to resum the Furry-picture α expansion in order to take quantum radiation reaction and spin transition into account in the nonlinear trident process in (pulsed) plane-wave background fields. The results are therefore nontrivial functions of both the background field strength, eE, and the coupling to the quantized photon field, α=e2/4π. The effective expansion parameter, T, is α times eE/mω≫1, which makes higher orders in α important. We show that they can change the sign of the spin-dependent part already at T<1, which will be experimentally accessible. We also present a new resummation method that essentially does to a convergent series what Borel-Padé resummation does to an asymptotic series.