Making sense of Dirac's rotation operator in "General Theory of Relativity"

[Kostik]

Actually, both the Weinberg and Dirac helicities are eigenvalues, just those of two distinct (but related) eigensystems.
In terms of Dirac's $R_{DI}$, Weinberg's helicities are the eigenvalues $\omega_i$ of the eigensystem:$$iR_{DI}\left(w_{i}\right)\equiv i\left(R_{DI}w_{i}+w_{i}R_{DI}^{T}\right)=\omega_{i}w_{i}\tag{W}$$where the $w_i$ are the eigentensors of $iR_{DI}$. After some linear algebra, the 4 solutions of eigensystem $\left(\text{W}\right)$ are:$$
w_{+}=\left(\begin{array}{cc}
-1 & i\\
i & 1
\end{array}\right),\:w_{-}=\left(\begin{array}{cc}
-1 & -i\\
-i & 1
\end{array}\right),\:w_{a}=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right),\:w_{b}=\left(\begin{array}{cc}
0 & -1\\
1 & 0
\end{array}\right)
$$along with their associated eigenvalues:$$\omega_{\pm}=\pm2,\:\omega_{a}=\omega_{b}=0$$Using this Weinberg eigentensor basis, we can decompose a 2D general (not necessarily symmetric) real rank-2 tensor $u$ as:$$
u\equiv\left(\begin{array}{cc}
u_{11} & u_{12}\\
u_{21} & u_{22}
\end{array}\right)=u_{2}^{+}w_{+}+u_{2}^{-}w_{-}+u_{0}^{a}w_{a}+u_{0}^{b}w_{b}=\left(\begin{array}{cc}
u_{0}^{a}-u_{2}^{+}-u_{2}^{-} & -u_{0}^{b}+i\left(u_{2}^{+}-u_{2}^{-}\right)\\
u_{0}^{b}+i\left(u_{2}^{+}-u_{2}^{-}\right) & u_{0}^{a}+u_{2}^{+}+u_{2}^{-}
\end{array}\right)
$$Solving for the mode coefficients of the eigentensors in terms of the matrix entries gives:$$u_{2}^{\pm}=\frac{1}{4}\left(u_{22}-u_{11}\mp i\left(u_{12}+u_{21}\right)\right),\:u_{0}^{a}=\frac{1}{2}\left(u_{11}+u_{22}\right),\:u_{0}^{b}=\frac{1}{2}\left(u_{21}-u_{12}\right)$$So we have the following table of eigenmode coefficients and associated helicity eigenvalues:
$$
\begin{array}{|c|c|}
\hline \text{Mode} & \text{Weinberg helicity } h=\omega\\
\hline u_{22}-u_{11}\mp i\left(u_{12}+u_{21}\right) & \pm2\\
\hline u_{11}+u_{22} & 0\\
\hline u_{21}-u_{12} & 0\\
\hline
\end{array}
$$As you note, Weinberg's helicity-2 modes are necessarily complex.
Dirac, on the other hand, considers the eigensystem constructed from the square of $iR_{DI}$, solves to find that system's eigenvalues $\delta_i$ and then defines $\pm\sqrt{\delta}$ to be the helicity of a mode. Specifically, the Dirac eigensystem is:$$iR_{DI}\left(iR_{DI}\left(d_{i}\right)\right)=\delta_{i}d_{i}\tag{D}$$with the 4 solutions:$$
d_{1} =\left(\begin{array}{cc}
-1 & 0\\
0 & 1
\end{array}\right),\:d_{2}=\left(\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right),\:d_{3}=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right),\:d_{4}=\left(\begin{array}{cc}
0 & -1\\
1 & 0
\end{array}\right)
$$$$\delta_{1}=4,\:\delta_{2}=4,\:\delta_{3}=\delta_{4}=0$$Decomposing $u$ in this Dirac basis yields:$$
u\equiv\left(\begin{array}{cc}
u_{11} & u_{12}\\
u_{21} & u_{22}
\end{array}\right)=u_{1}d_{1}+u_{2}d_{2}+u_{3}d_{3}+u_{4}d_{4}=\left(\begin{array}{cc}
-u_{1}+u_{3} & u_{2}-u_{4}\\
u_{2}+u_{4} & u_{1}+u_{3}
\end{array}\right)
$$so that:$$u_{1}=\frac{1}{2}\left(u_{22}-u_{11}\right),\:u_{2}=\frac{1}{2}\left(u_{12}+u_{21}\right),\:u_{3}=\frac{1}{2}\left(u_{11}+u_{22}\right),\:u_{4}=\frac{1}{2}\left(u_{21}-u_{12}\right)$$The helicity table for Dirac eigenmodes is therefore:
$$
\begin{array}{|c|c|}
\hline\text{Mode} & \text{Dirac helicity } h=\pm\sqrt{\delta}\\
\hline u_{22}-u_{11} & \pm2\\
\hline u_{12}+u_{21} & \pm2\\
\hline u_{11}+u_{22} & 0\\
\hline u_{21}-u_{12} & 0\\
\hline
\end{array}
$$All of Dirac's modes are real.
Weinberg's approach is natural for describing complex gravitational plane waves while Dirac's definition has the virtue of yielding strictly real eigenmodes for a given helicity. In my opinion, which is the "right" version of eigenmodes and helicities comes down to a matter of convenience and/or taste.

Just a last note, while you compute the four eigenbasis matrices $d_i \, (i=1,2,3,4)$ of the operator $(iR)^2$ which have eigenvalues $4, 0$, you do not immediately see the eigenbasis matrices $w_i$ of $iR$ which have eigenvalues $\pm 2, 0$. These were, of course, obtained previously and directly by solving the eigenvalue equation with the operator $iR$.