Are profinite groups of cardinality $|\mathbb{R}|$ determined by their finite quotients?

Without the extra finiteness conditions, but also not relying on any set-theoretic assumptions, take $G = \widehat{\mathbb{Z}}^{\mathbb{N}}$ and $H = G \times A$ for a nontrivial finite abelian group $A$. Both $G$ and $H$ have every possible finite abelian quotient, but only $H$ has torsion.