Matrix rows or columns are traditionally listed under $(x,y,z)$ order.
Cyclically change the pairs under consideration i.e $(x,y)\to(y,z)\to(z,x)$. The pairs $(x,y)$ and $(y,z)$ show up in the same order in the matrix but the $(z,x)$ shows up in reverse in the matrix. That is the cause of apparent discrepancy but really there is no discrepancy.
For example write
$x'=x\cos \alpha - y \sin \alpha$
$y'=x\sin \alpha + y \cos \alpha$
now change $(x,y)\to(y,z)\to(z,x)$ and $\alpha\to \beta \to \gamma$ and write the three matrices to see how $(z,x)$ part gets flipped.
Edit:
If you want them to look alike then give up the matrix notation and instead write
$y'=y\cos \beta - z \sin \beta$
$z'=y\sin \beta + z \cos \beta$
And
$z'=z\cos \gamma - x \sin \gamma$
$x'=z\sin \gamma + x \cos \gamma$
In each instance if you try to write $\left[ \matrix{ x' \cr y' \cr z'}\right]$ in terms of $\left[ \matrix{ x \cr y \cr z}\right]$ you will see that the mystery goes away.
After a lot of searching, I have finally found a reference that has what I needed. Although it doesn't appear in a 'Quaternion to Euler' or 'Quaternion to Tait-Bryan' google search, when I gave up and started looking for 'Quaternion to Axis-Angle' with the intention of going through that representation as an intermediate step, I came across the following wonderful document:
Technical Concepts
Orientation, Rotation, Velocity and Acceleration,
and the SRM
Version 2.0, 20 June 2008
Author: Paul Berner, PhD
Contributors: Ralph Toms, PhD, Kevin Trott, Farid Mamaghani, David Shen,
Craig Rollins, Edward Powell, PhD
http://www.sedris.org/wg8home/Documents/WG80485.pdf
It covers a lot of the formalisms, but most importantly, shows derivations and solutions for 3-1-3 and 3-2-1 Euler angle representation. It also seems to cover inter-conversion between pretty much every other rotation representation I'm aware of, and so I would also recommend it as a good general reference.
Oh, and the actual solution for a 3-2-1 ($z-y-x$) Tait-Bryan rotation convention from that reference:
$$
\phi = \operatorname{arctan2}\left(q_2 q_3 + q_0 q_1,\frac{1}{2}-(q_1^2 + q_2^2)\right) \\
\theta = \arcsin(-2(q_1 q_3 - q_0 q_2)) \\
\psi = \operatorname{arctan2}\left(q_1 q_2 + q_0 q_3,\frac{1}{2}-(q_2^2 + q_3^2)\right)
$$
Note that the gimbal-lock situation occurs when $2(q_1 q_3 + q_0 q_2) = \pm1$ (which gives a $\theta$ of $\pm \frac{\pi}{2} $), so it can be clearly identified before you attempt to evaluate $\phi$ and $\psi$.
(Convention for arctan2 is $\operatorname{arctan2}(y, x)$, as hinted on page 3.)
Best Answer
The relationship between the two frames as you’ve drawn them is simple: rotate by an angle of $\pi/2$ radians counterclockwise around the z axis, then reflect through the plane perpendicular to y.
There isn’t going to be any elegance about it with Euler angles. Someone has already done the repulsive conversion from unreal to 3x3 matrix or to quaternions for you, so find that. Just pick one and stick with it.
By “handed” it sounds like you mean the choice of sign when expressing a rotation around an axis. If that bothering you then obviously you can swap direction by swapping signs. But like I said, if it’s coming out of unreal someone probably already has written a conversion to a normal right handed coordinate system. (That’s what I think of as handedness: an orientation assigned to a coordinate system, not a type of rotation.)
Performing the rotation and reflection is trivial using matrices or quaternions. Rotations by quaternions are of course famous. Reflections might be lesser known but they are pretty simple to describe: take the unit normal $n$ to your plane expressed a s a quaternion: then $x\mapsto nxn $ is the reflection.