This question can be cast in the language of branching random walks: at each (discrete) time, a particle splits with probability $p$ and each of the children moves $\log \epsilon$; if it did not split, it moves $\log 2$. You are asking about the largest particle moving to infinity, i.e. the velocity being positive (essentially). This fits the model described e.g. in Zhan Shi's, specifically Theorem 2.3 there. Let $\psi(t)= \log (2p \epsilon^t+q 2^t)$, then $v=\inf \psi(t)/t$, and you are asking whether $v>0$ or not. This is the sought after criterion.
The link to Shi's lecture notes is: https://link.springer.com/book/10.1007/978-3-319-25372-5