This is long for a comment
Under the assumption that we can cancel, and multiply by inverses, let's simplify $ \color{red}{x_A(\phi_A)}x_\beta(\phi(\beta))x_B(\phi_B)x_\beta(t)-\color{red}{x_\beta(t)}\stackrel{?}{=} \color{red}{x_A(\phi_A)}x_\beta(\phi(\beta)+t)[x_B(\phi_B),x_\beta(t)]-\color{red}{x_\beta(t)} $ We get $ x_\beta(\phi(\beta))x_B(\phi_B)x_\beta(t)\stackrel{?}{=} x_\beta(\phi(\beta)+t)[x_B(\phi_B),x_{\beta} (t)] $ The proof say by "By relation (3.6)", and you have $ x_{\alpha}(a) x_{\alpha}(b) = x_{\alpha}(a+b) \tag{3.6}$
Here is a question for you. According to (3.5), does we have $ x_B(\phi_B) x_{\beta}(t) = x_{\beta}(t)[x_B(\phi_B),x_\beta(t)]??$ I'm guessing so. Then by (3.6) we have $ x_\beta(\phi(\beta))x_B(\phi_B)x_\beta(t) \stackrel{?}{=} \color{red}{x_\beta(\phi(\beta))x_\beta(t)} [x_B(\phi_B),x_\beta(t)]\\ \stackrel{?}{=} \color{red}{x_{\beta}(\phi(\beta)+t)} [x_B(\phi_B),x_\beta(t)]$ So you get what you want.
But again.. warning: I only skimmed few pages of the paper and all of this are just guessing.