This type of question (separating classes by representations) is natural throughout representation theory of various different flavors. In particular, the systematic treatment of (finite dimensional) irreducible representations of connected compact Lie groups leads to a straightforward affirmative answer: see for instance the textbook Representations of Compact Lie Groups by Brocker and tom Dieck (Springer GTM 98). I could track down specific references in this book, but it's important to realize that what you want is already proved in such sources.
Briefly, the compact Lie group case involves some structure theory: all maximal tori are conjugate and every element is conjugate to some element in your fixed $T$. Moreover, two elements of $T$ are conjugate iff they are conjugate under the Weyl group $W = N_G(T)/T$. Then one has to pin down the class functions on $G$ (invariant on conjugacy classes) by identifying them with $W$-invariant functions on $T$. (Here the category of groups determines what kind of functions are relevant.) A key fact is that characters of irreducible (necessarily finite dimensional) representations generate the algebra of class functions on $G$, from which you pass to the isomorphic algebra of $W$-invariant functions on $T$. These ingredients are standard but not trivial to develop. As a refinement, when $G$ is semisimple and simply connected, one sees that classes in $G$ are already separated just by the values of the finitely many fundamental characters.
To get more perspective on these ideas, it's worthwhile to look at the closely parallel treatment of a connected semisimple algebraic group (over an algebraically closed field of any characteristic): work of Steinberg and others. Here virtually the same results can be proved, with the important difference that not all elements of $G$ are semisimple (and indeed, characters of representations fail to distinguish elements from their semisimple parts). See Steinberg's old Tata lecture notes or Chapter 3 of my book on conjugacy classes.
