This is a simplified version of the original question asked, it seems that some parts are unnecessary and only confuse. I will therefore here ask a related, but I believe simpler, version of the question. Let $0 < \lambda < 1$ and let $$S_n = X_1 + \dots + X_n$$ be a random walk with i.i.d. increments $X_i$ with law $P(X_1 =0) = P(X_1 =1 ) = 1/2$. Now, consider $$ E \left( \frac{\lambda^{S_n}} {\sum_{k=0}^{n-1} \lambda^{S_k}} \right).$$ The task is to find a sequence $f(n)$ (possibly random) such that $$ E \left( \frac{\lambda^{S_n}} {\sum_{k=0}^{n-1} \lambda^{S_k}} \right) \Big/ {f(n)} \to 1 $$ and thus, find the growth behavior of the expectation at question. Note that $\sum_{k=0}^{n-1} \lambda^{S_k} \to L:= \sum_{k=0}^{\infty} \lambda^{S_k}$ a.s.
Here the question from before (first part deleted):
[...] Then, I am interested in general, what is the asymptotic growth behavior of $$ E \left( \frac{\lambda^{S_n} (1- \lambda^{S_n})}{\lambda^{S_n} + \sum_{k=0}^n \lambda^{S_k}} \right) ?$$