The easiest method might be just to prove that both expressions define the same language as the automaton you started with. This is quite straightforward, and you have done half of the work already.
Specifically, $R+SU^*T$ is the expression for “go from the start state to the start state with all intermediate states being the other state”, and $SU^*$ the expression for “go from the start state to the final state without ever going back”, yielding $(R+SU^*T)^*SU^*$. I believe this is exactly what you call the “$R_{ij}^{(k)}$ method”—you just use a different implicit ordering of the states. (The general method is described here.)