Which word could I use for the pumping lemma?
So it has to be something like this $w=b{a}^{2m+2}b^mc^m$.


$b{a}^{2m+2}b^mc^m$ is not the general form for $w=bxc^m\in S$. If $x$ must be in the form of $a^*b^*$, $w$ will be something like $b{a}^{m+n+k+2}b^nc^m$ for some $n,k\ge0$. Just for completeness, $w$ could be like $baba^{m+1}c^m$ or  $baba^{m+1}a^kb^kc^m$ or many other forms. Of course, it is totally fine if you just meant to choose one word so as to apply the pumping lemma.



Can you check word $ba^{p+2}c^{p}$, assuming the pumping length is $p$? This word is about as simple as you could get.



Exercise 1. Assuming the pumping length is $p$, can we pump $ba^{p+3}bc^p\in S$?


Exercise 2. Assuming the pumping length is $p$, can we pump $ba^{2p+2}b^pc^p\in S$?