We present a measurement of the transverse single-spin asymmetry for π⁰ and η mesons in p↑ + p collisions in the pseudorapidity range IηI < 0.35 and at a center-of-mass energy of 200 GeV with the PHENIX detector at the Relativistic Heavy Ion Collider. In comparison with previous measurements in this kinematic region, these results have factor-of-3 -smaller uncertainties. As hadrons, π⁰ and η mesons are sensitive to both initial- and final-state nonperturbative effects for a mix of parton flavors. Comparisons of the differences in their transverse single-spin asymmetries have the potential to disentangle the possible effects of strangeness, isospin, or mass. These results can constrain the twist-3 trigluon collinear correlation function as well as the gluon Sivers function.