An ultrarelativistic electron beam passing through an intense laser pulse emits radiation around its direction of propagation into a characteristic angular profile. Here, we show that measurement of the variances of this profile in the planes parallel and perpendicular to the laser polarization, and the mean initial and final energies of the electron beam, allows the intensity of the laser pulse to be inferred in a way that is independent of the model of the electron dynamics. The method presented applies whether radiation reaction is important or not, and whether it is classical or quantum in nature, with an accuracy of a few percent across 3 orders of magnitude in intensity. It is tolerant of electron beams with a broad energy spread and finite divergence. In laser-electron-beam collision experiments, where spatiotemporal fluctuations cause the alignment of the beams to vary from shot to shot, this permits inference of the laser intensity at the collision point, thereby facilitating comparisons between theoretical calculations and experimental data.