Are there convenient methods/tricks to make calculations with non-independent terms? (two examples here in particular)

In a recent question it came out that I needed to calculate the sample distribution of $dfrac{Cov(XY,X)}{Var(X)}$ where the distributions of $X$ and $Y$ are known.
By this, I mean the law of $$dfrac{left(sum_{i=1}^NX_i^2Y_iright)-dfrac{sum_{j=1}^NX_jY_jsum_{k=1}^NX_k}{N}}{left(sum_{i=1}^NX_i^2right)-dfrac{left(sum_{j=1}^NX_jright)^2}{N}}$$

where $N$ is fixed and the $X_i$ are i.i.d., say $mathcal N(0,1)$, and the $Y_i$ are i.i.d. as well, say $mathcal N(mu,1)$.

I thought that I could use the CLT, but it turns out that it’s more complicated than that.

What I can do is say that since the $X_i$ have mean $0$, I can wave my hands and argue that the second part of the numerator, $dfrac{sum_{j=1}^NX_jY_jsum_{k=1}^NX_k}{N}$, is reasonably close to $0$, and so is the second part of the denominator. I’m left with $${dfrac{sum_{i=1}^NX_i^2Y_i}{N}}left/, {dfrac{sum_{i=1}^NX_i^2}{N}}right.$$
where now the CLT can be applied and the numerator is distributed as $mathcal N left(mu, frac{3+2mu^2}{sqrt N}right)$ while the denominator is distributed as $mathcal N left(1, frac{2}{sqrt N}right)$.

Again (that’s the second example as promised in the title), I would love to be able to say that the denominator is close to $1$ and forget about it, but that’s even less rigorous than it seems since the numerator and denominator are not independent.

Are there known methods to carry this out rigorously or at least control the error induced by such approximations?