Previous studies of the reaction ³He+⁴He→⁷Be+γ have mainly focused on providing the best central value and error bar for the S factor at solar energies. Experimental measurements of this capture reaction at higher energies, the ³He – ⁴He scattering phase shifts, as well as properties of ⁷Be and its excited state, have been used to constrain the theoretical models employed for this purpose. Here we show that much more information than was previously appreciated can be extracted from angle-integrated capture data alone. We use the next-to-leading-order (NLO) amplitude in an effective field theory (EFT) for ³He+⁴He→⁷Be+γ to perform the extrapolation. At this order the EFT describes the capture process using an s-wave scattering length and effective range, the asymptotic properties of ⁷Be and its excited state, and short-distance contributions to the E1 capture amplitude. We extract the multi-dimensional posterior of all these parameters via a Bayesian analysis that uses capture data below 2 MeV. We find that properties of the ⁷Be ground and excited states are well constrained. The total S factor keV b, while the branching ratio for excited- to ground-state capture at zero energy, , both at 68% degree of belief. This S(0) is broadly consistent with other recent evaluations, and agrees with the previously recommended value S(0)=0.56±0.03keV b, but has a smaller error bar. We also find significant constraints on ³He – ⁴He scattering parameters, and we obtain constraints on the angular distribution of capture gamma rays, which is important for interpreting experiments. The path forward for this reaction seems to lie with better measurements of the scattering phase shifts and S(E)’s angular dependence away from zero energy, together with better understanding of the asymptotic normalization coefficients of the ⁷Be bound states’ wave functions. Data on these could further reduce the uncertainty on S(0).