In this paper, the Hartemann-Kerman approach is numerically examined with the implementation of Bessel pulses that dramatically changes the morphology of yielded radiation. For this end it is used the covariant Bessel pulse given by J (Φ/Δ) with J the integer-order of Bessel function whereas Φ and Δ the invariant product and the pulse width respectively. In concrete this paper is focused at the emitted radiation by a free electron that depends on the pulse amplitude as well as the normalized Doppler-shifted frequency. The numerical analysis yields that a fully nonlinear spectrum that to some extent can be interpreted as of GeV electron to interact simultaneously with a laser power of 1.36 keV and a peak intensity of order of 1000 W/cm2. Therefore, the usage of Bessel-like pulses would reveal the departure of a classical view to one based at electron-photon interaction as dictated by the quantum mechanics rules.