The distribution of $T$ is indeed asymptotically normal, by virtue of any one of the many central theorems for stationary weakly dependent (here, even $2$-independent) random variables (r.v.'s). See e.g. Theorem 0 in Bradley (due to Ibragimov), the conditions of which are easily verified in your case.
An idea of how to get such results goes back to Bernstein and is as follows: Partition the sequence of weakly dependent r.v.'s into an alternating sequence of longer and shorter blocks, such that each long block is followed a short one, but the long blocks are still much shorter than the number (say $n$, $n\to\infty$) of the weakly dependent r.v.'s, so that the number of the long blocks is growing to infinity with $n$. Then the contribution of the short blocks will be comparatively small, whereas the long blocks will be nearly independent. In your case, you can take the short blocks just of length $1$, an then the long blocks, of length much greater than $1$ but much less than $n$, will be exactly independent.