Countable discrete extensions of compact lines

We consider a separable compact line $K$ and its extension $L$ consisting of $K$ and countably many isolated points. The main object of study is the existence of a bounded extension operator $E: C(K)\to C(L)$. We show that if such an operator exists, then there is one for which $\|E\|$ is an odd natural number. We prove that if the topological weight of $K$ is greater than or equal to the least cardinality of a set $X\subseteq [0,1]$ that cannot be covered by a sequence of closed sets of measure zero, then there is an extension $L$ of $K$ admitting no bounded extension operator.