Concatenation among different language types
The language $L'$ contains all possible strings, and so it is in particular regular. This is because the empty word is in $L$, and so $L\cdot \{0,1\}^* \cdot L^R$ contains $\{0,1\}^*$. If you "proved" that it's not regular using the pumping lemma, you have made a mistake applying it. Try to find it; this will lead to better understanding of the pumping lemma.