I would be curious to know how you're applying the renewal theorem here. For me, the renewal theorem tells that the "excursions" outside a given state are i.i.d. (this is just Markov property actually), thus if is the time of -th visit of state (with ) we have . Since , I would conjecture that the limit is , but I have no formal proof. The max comes from the usual (if ), and . The step from conjecture to formal proof would go through replacing by the renewal time that is closest to .
Another approach would be using matrices. Indeed, one can read the quantity on a matrix product... Denote by the matrix obtained from by putting 0's on the first line and first column: for any . Then . Indeed, the probability of paths that don't go through state 1 can be computed with with no change, and paths that go through state 1 lead to a term equal to 0 in the matrix product. It would also be possible to put 0's only on the first line and consider (paths of length n+1 that don't exit from x at any time correspond to paths of length n that don't visit x at any time).
Then is the largest eigenvalue of (in modulus)... its logarithm would therefore be the answer to the question. This resumes to an algebraic problem... that doesn't seem trivial? (but is numerically simple)