I took some time, but I hope it still helps. Coming back to your problem I realized that the norm constraint is not convex (you should have the opposite inequality for convexity).

Assuming you have the opposite inequality, you can write the problem in standard form:

\begin{eqnarray*}
\min & Tr(XA) \\
\mbox{s.t.} & Tr(X A^{\prime})+t &\geq a \\
            & vec(X)^HA^{\prime\prime} &= z \\
            & (z,t)&\in L^{M+1}\\
            & X &\succeq 0
\end{eqnarray*}

where $L^{M+1}$ is the Lorentz cone in $M+1$ variables. The nice thing about barrier functions is that they (and their complexity) are additive (see http://www2.isye.gatech.edu/~nemirovs/Lect_ModConvOpt.pdf, page 276), so the complexity of interior point methods would be the sum of the barrier parameter of the PSD cone + the barrier parameter of the Lorentz cone.

Of course, all this provided you have a convex program.