Let me answer my own question by contructing, for $p>2$ not an even
integer (say $2k<p<2k+2$), a matrix $A$ such that $\|A\|_p> \|A'\|_p$. In fact I construct a family of matrices $A_n \in M_n(\mathbb C)$ such that $\|A_n\|_p > \|A_n'\|_p$ whenever $n-k$ is an even positive integer, and $\|A_n\|_p < \|A'_n\|_p$ if $n-k$ is an odd positive integer$.

This will imply that $\|A\|_p \leq \|A'\|_p$ for all $n$ and all $A \in M_n$ if and only if $p$ is an even integer or $p=\infty$. For other values of $p$, these two quantities are not comparable, and if we go the the Schatten classes (ie we allow infinite matrices), there is no implication between the properties $\|A\|_p<\infty$ and $\|A'\|_p<\infty$.

Here is the construction. For every integer $n$, consider $S_n \in M_n(\mathbb C)$ to be the matrix of a cyclic permutation of $\{{1,\dots,n\}}$ in which one of the $1$'s is replaced by a $-1$. Take $A_n=Id+S_n$, so that $A_n'=Id+S_n'$. I only sketch the proof that $A_n$ works, since I might be the only one interested ;).

There are two independent claims:

> *Claim 1*: the function $p \mapsto \|A_n\|_p^p - \|A_n'\|_p^p$ has at most $n-1$ zeros (counting multiplicities).



> *Claim 2*: $\|A_n\|_p = \|A_n'\|_p$ for all even integers $2 \leq p \leq 2n-2$.

These two claims together imply that $\|A_n\|_p^p \neq \|A_n'\|_p^p$ is non zero outside of $\{{2,4,\dots,2n-2\}}$ and changes signs at each of these values of $p$. Since it is negative for $p=2n$, we have the announced properties.

The second claim is easier. A first observation is that $Tr(S_n^k) = Tr(S'_n^k)$ for every $k$ with $-n+1 \leq k \leq n-1$: $k=0$ is obvious, and if $k \neq 0$ both matrices have a zero diagonal. Hence, since $A_n^* A_n = 2+S_n +S_n^*$ and  $A'_n^* A'_n = 2+ S'_n +S'_n^*$ we have $\|A_n\|_p = \|A_n'\|_p$ for all even integers $p \leq 2n-2$.

To prove the first claim, first notice that the eigenvalues of $S'_n$ (resp. $S_n$) are $\lambda_k=\exp(2ki\pi/n)$, $k=1\dots n$ (resp. $ \mu_k=\exp( (2k+1) i \pi/n)$, $k=1\dots n$). Thus the singular values of $A'_n$ (resp. $A_n$) are $|1+\lambda_k|$ (resp. $|1+\mu_k|$). In particular if $N(B)$ denotes the number of distinct non-zero singular values of a matrix $B$, we have $N(A_n)+N(A'_n)=n$. Hence $\|A_n\|_p^p - \|A_n'\|_p^p$ can be written in the form $\sum_{j=1}^n \alpha_j e^{\beta_j p}$, and such a function cannot have more than $n-1$ zeros unless it is identically zero.