Central Limit Theorem via Fixed Point Theorem and Entropy

I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is:

1. The average $\bar{X} = \frac{1}{n} \sum^n_{i=1}X_i$ of iid continuous random variables can be described by its probability density function. The PDF of a sum of continuous random variables is the convolution of the two individual PDFs.
2. Find an appropriate operator that uses convolution. If it is a contracting map on the space of PDFs of continuous random variables, then the Banach Fixed Point Theorem says there exists a unique fixed point.
3. For a fixed mean $\mu$ and variance $\sigma^2$, the normal distribution $N(\mu,\sigma^2)$ has maximum entropy among all random variables with this mean and variance. On the other hand, convolution never decreases entropy, only keeps it the same or increases it.
4. Use this monotonicity fact to prove the operator is a contracting map on this subspace of random variables of fixed mean and variance. Note that the appropriateness of the operator should be that mean and variance are preserved. For simplicity, we might just assume $\mu=0$ and have the operator be $T_n(f) = (f*...*f)/\sqrt{n}$ where we convolve $n$ times. This map should only ever increase entropy but also, there a maximum entropy belonging to the normal distribution. So prove the maximizer of entropy is the unique fixed point.

Questions: Does this rough outline make sense and how would it be made rigorous? I'm wanting to use entropy as the means for proving the map is contracting. One issue of confusion for me is that we wouldn't want to reapply $T_n$ but rather allow $n$ to grow larger. But the setting of the fixed point theorem is that repeated iterations of a contracting map should converge to the fixed point. So we could just set $n=2$ and $k$ iterations will have $2^k$ convolutions.

A version of this question has been asked here but I didn't really follow the ideas. In particular, I wasn't able to find the references mentioned, at least not in English translation or free online. One of the answers mentions a stable manifold of the Gaussian fixed point and that is sort of the picture I have in mind: the stable manifold referenced there is the subspace on which I want to apply the convolution map and fixed point theorem.