Well, the constant C in Lemma 1 is given explicitly as something which can probably be bounded by 100. The constant C in Lemma 4 is stated to be 4. Now, in the proof of Theorem 4, Case I, they obtain the absolute constant bound C(2e)^(1+1/q), with C the constant from Lemma 4. For Case II, the constant in the first terms can be bounded by 2 times the constant from Lemma 1, so 200. The constant bounding the second term is just the constant from Lemma 4. So, up to bounding the Beta function by q, you should be able to take C = 800 in your question. It is not optimal, but not far from the truth.

Maybe I’m a bit late, but the constants in Lemmas 1 and 4 of the original Carbery Wright paper seem to be explicit enough to get an expression for the cosnstant.

Not in general. But at least when the X_i are Rademacher, there is no convergence in Total variation. In this case, F_n is always supported on a lattice, which has Gaussian measure 0.

The problem is that we really need $X$ to commute with $\mathbb{E}[X]$, which is generally not true.

The thing is, it's not obvious how to 'smear' a discrete random variable into one which a Poincare inequality. Satisfying such inequalities, means the density cannot be too 'bumpy', in the sense that it cannot be 0 at any interval (otherwise just choose $f$ to be a bump function on the complement of the interval).

Well, $X$ is fixed. The sequence is $a_iX$ in essence. Sp, they are not identical, but up to re-scaling all elements of the sequence are like $X$. A random variable satisfies Poincare inequality if there is a positive constant $c$ such that for every (differentiable) $f$, $c\mathrm{Var}(f(X)) \leq E[f'(x)]^2$. Basically, what is important here, that $X$ cannot be supported on 'atoms'.