This is proved for real non-negative $k$ and $\ell$ in Chapter 2 of my thesis (I'm currently making minor corrections to it, and a version of this Chapter should appear on the arXiv before too long).
One adapts the lower bounds method of Radziwi{\l}{\l}-Soundararajan (https://arxiv.org/abs/1202.1351v1), or more relevantly for the Riemann zeta function Heap-Soundararajan's work (https://arxiv.org/abs/2007.13154).
Very briefly, with a generalisation of the Hölder's inequality in Heap--Soundararajan's paper one has (assuming that $k\geq \ell$):
\begin{equation}\label{eq:cts-lb-holder-1}
\begin{split}
\Big|\int_T^{2T}\zeta^{(m)}(\tfrac{1}{2}+it)&\zeta^{(n)}(\tfrac{1}{2}+it)|\mathcal{N}(\tfrac{1}{2}+it)|^2 \ d t\Big|\\
\ll &
\Big(\int_T^{2T}|\zeta^{(m)}(\tfrac{1}{2}+it)|^{2k}|\zeta^{(n)}(\tfrac{1}{2}+it)|^{2\ell}\ d t\Big)^{1/(2u(k+\ell))}\\
&\quad \times\Big(\int_T^{2T}|\mathcal{N}(\tfrac{1}{2}+it)|^{2+2/(k+\ell-1)}\ d t\Big)^{(k+\ell-1)/(2u(k+\ell))}\\
&\quad \times{\Big(\int_T^{2T}|\zeta^{(m)}(\tfrac{1}{2}+it)|^2|\mathcal{N}(\tfrac{1}{2}+it)|^2\ d t\Big).}^{1-1/(2u)}
\end{split}
\end{equation}
where $u=\ell/(k+\ell)$ and the notation $\mathcal{N}(s,\cdot)$ is the same as in Heap-Soundararajan.
The idea is that the $\mathcal{N}$ is a Dirichlet polynomial approximation to zeta that ensures that Hölder's inequality is sharp-up-to-constant and each factor has the same order of magnitude.
The fractional Dirichlet polynomial moment is essentially the same as Heap-Soundararajan's and the twisted moment calculations can be done by differentiating the usual formulae for the twisted second and fourth moments of zeta.
By the way, is there a particular application that you were thinking of for this result?
