Models with radiative symmetry breaking typically feature strongly supercooled first-order phase transitions, which result in an observable stochastic gravitational wave background. In this work, we analyse the role of higher-order thermal corrections for these transitions, applying high-temperature dimensional reduction to a theory with dimensional transmutation. In particular, we study to what extent high-temperature effective field theories (3D EFT) can be used. We find that despite significant supercooling down from the critical temperature, the high-temperature expansion for the bubble nucleation rate can be applied using the 3D EFT framework, and we point out challenges in the EFT description. We compare our findings to previous studies and find that the next-to-leading order corrections obtained in this work have a significant effect on the predictions for GW observables, motivating a further exploration of higher-order thermal effects.