Regular language is closed given transposition of rightmost character to leftmost
This answer assumes that the transposed language never contains the empty string.
Let $L$ be a regular language, say accepted by a DFA with states $Q$, initial state $q_0$, accepting states $F$, and transition function $\delta$.
We construct a new DFA with states $Q' = q'_0 \cup (Q \times \Sigma)$. The initial state of the new DFA is $q'_0$. The transition function is defined as follows: $\delta'(q'_0,\sigma) = (q_0,\sigma)$, and $\delta'((q,\sigma),\tau) = (\delta(q,\tau),\sigma)$. Finally, the set of accepting states $F'$ contains all states $(q,\sigma)$ such that $\delta(q,\sigma) \in F$.
The new DFA reads the first symbol $\sigma$ and remembers it. It then simulates the original DFA, accepting a word if adding $\sigma$ would result in the original DFA accepting it.