Local solutions to the 3D stochastic quantisation equations of Yang-Mills-Higgs were constructed in Chandra (Invent Math 237:541-696, 2024), and it was shown that, in the limit of smooth mollifications, there exists a mass renormalisation of the Yang-Mills field such that the solution is gauge covariant. In this paper we prove the uniqueness of the mass renormalisation that leads to gauge covariant solutions. This strengthens the main result of Chandra (Invent Math 237:541-696, 2024), and is potentially important for the identification of the limit of other approximations, such as lattice dynamics. Our proof relies on systematic short-time expansions of singular stochastic PDEs and of regularised Wilson loops. We also strengthen the recently introduced state spaces of Cao (Comm Part Diff Equ 48:209-251, 2023); Cao (Comm Math Phys 405:3, 2024); Chandra (Invent Math 237:541-696, 2024) to allow for finer control on line integrals appearing in expansions of Wilson loops.
