@whuber always a pleasure to read your answers. Cheers.

Neat! Thanks. So basically my expression is equivalent to a particular sum of harmonic numbers?

Also, I believe there is a small typo in the distribution of $Z$ where you have $-t \le t \le 1$ it should be -1, but it won't let me change it.

That - at least as far as I follow, says $UV, U+V$ are dependent, but the first part of the proof shows that $XY, X+Y$ have 0 covariance (which I know doesn't necessarily imply independence).

Does the first argument show that XY and X+Y are independent?

I have done this exercise when $X,Y$ are standard uniform, but have found this to be trickier for whatever reason.

I have done this, but I keep getting an undefined result when $x < 0$. Can you expand?

I'm concerned that this method is based on taking a linear expansion - valid, provided that higher order terms can be neglected. But truthfully it only seems you can neglect this other term if $(g')^2 \gg g g''$ otherwise that term isn't really a "higher" order term. What I am trying to point out, I guess not really clearly is that this other term is ALSO a second order central moment term.

You haven't really addressed the question. The question is why is the second-order term neglected when it seems to not only be the same magnitude, but precisely cancel in the case of the Poisson distribution. Often series approximations are truncated on the basis that higher order terms can be neglected. This seems to not be the case here.