Gravity-induced non-Gaussianity in the large-scale structure of the Universe, characterised by higher-order statistics such as the bispectrum (three-point cumulant), is expected to contain rich cosmological information. A measurement of the bispectrum will not only improve the cosmological constraints, but also give us the possibility to probe gravity on cosmological scales. We present a framework to numerically calculate the one-loop matter bispectrum based on standard perturbation theory (SPT). This approach allows general modifications to the standard ΛCDM model to be easily implemented. We demonstrate the performance of the bispectrum calculation in three representative cases, namely the Vainshtein-screened Dvali-Gabadadze-Porrati (DGP) model, the chameleon-screened Hu-Sawicki f(R) model and the phenomenological dark scattering (DS) momentum-exchange model. We compare with measured results from a set of cosmological N-body simulations, and study in detail the impact of possible systematics arising from simplified or approximate treatments. We find that the one-loop bispectrum calculation offers significantly more information on general screening and momentum exchange effects than the leading-order bispectrum calculation. Further, the accuracy of the one-loop prediction is shown to be comparable to non-linear fitting formulas over a wide range of wavenumbers (k≤0.3 h/Mpc) even at lower redshifts, z ≤ 1. Finally, we discuss connecting the one-loop SPT approach to observations of the CMB lensing bispectrum and to the late time galaxy distribution.