I have been trying to prove this interesting proposition for a while. We have an extended class $\mathcal{K}$ function defined as follows:
Function $\alpha : (-c,d) \to (-c,\infty)$, that is strictly increasing and $\alpha(0) = 0$, is called an extended class function, denoted by $\mathcal{K}_e$.
Proposition: Given $\alpha \in \mathcal{K}_e$ and $b>0$, there exists $\gamma \in \mathcal{K}$ such that $\alpha (a - b) \leq \alpha(a) - \gamma (b)$.
It is easy to see that $\alpha (a - b) \leq \alpha(a)$, because $b$ is positive all the time. I am not able to go beyond this step.
This is one possible candidate for $\gamma$.
$\gamma (b) : = \inf_{a\in[-c,d)} \alpha(a) - \alpha(a - b)$.
It can be verified that $\gamma \in \mathcal{K}$. But this is true only for $d < \infty$.