X-ray irradiation heating of accretion discs in black hole X-ray binaries (BHXBs) plays a key role in regulating their outburst cycles. However, despite decades of theoretical and observational efforts, the physical mechanism(s) responsible for irradiating these discs remains largely unknown. We have built an observationally-based methodology to estimate the strength of irradiation of BHXB discs by combining multiwavelength X-ray and optical/IR data throughout transient outbursts. We apply this to $sim15$ yrs of activity in the Galactic BHXB GX339$-$4. Our findings suggest that the irradiation heating required by the optical data is large in this system. Direct illumination of the outer disc does not produce sufficient irradiation, but this should also produce a thermal-radiative wind which adds to the irradiation heating by scattering flux down onto the disc. However, analytic estimates of X-ray illumination from scattering in the wind is still not sufficient to produce the observed heating, even in combination with direct illumination. Either the analytic thermal-radiative wind models are underestimating the effect of the wind, or there are additional scattering mechanisms at work, such as magnetically-driven outflows, acting to increase the optical/IR flux. While wind-driven irradiation is likely a common feature among long-period BHXBs, fully understanding the driving mechanism(s) behind such a wind will require radiation-hydrodynamic simulations.