= Solution
The equivalence is \b[true]. Write $c_r(\eta)$ for the total size at level $r$ of the <core tower of a partition> $\eta$, and $q_r(\eta)$ for the corresponding total size in its <quotient tower of a partition>. We use the size relation and the <P-adic valuation of a symmetric-group character degree from the core tower>:
$$
|\eta|=\sum_{r\geq0}c_r(\eta)p^r,
\qquad
\nu_p(\chi^\eta(1))=\frac{\sum_{r\geq0}c_r(\eta)-d_p(|\eta|)}{p-1}.
$$
Here $d_p$ is the <base-p digit sum>. These imply the <character degree coprime to p from the core tower> criterion: if $|\eta|=\sum_r\alpha_rp^r$, then $p\nmid\chi^\eta(1)$ exactly when $c_r(\eta)=\alpha_r$ at every level. Indeed carrying $p$ units from one position to the next decreases the sum of coefficients by $p-1$; the valuation vanishes precisely when there are no carries.
For $k\geq1$, we also use <power-core truncation of a prime-core tower>: for $\gamma=C_{p^k}(\lambda)$, its core tower agrees with that of $\lambda$ below level $k$ and is empty at every level at least $k$. Moreover $q_k(\lambda)=w_{p^k}(\lambda)$, the <weight of a partition> for modulus $p^k$. To see the truncation, removal of a <rim hook> of length $p^k$ preserves the first $p$-core and, under the <abacus divisible-hook correspondence>, becomes removal of a $p^{k-1}$-hook in one quotient component. Iterating preserves all the lower cores; after all such removals, quotient level $k$ is empty.
If $p\nmid\chi^\lambda(1)$, the criterion gives $c_k(\lambda)=a$, $c_r(\lambda)=0$ for $r>k$, and the lower $c_r(\lambda)$ are the base-$p$ digits of $m$. It follows that $|\gamma|=m$, $p\nmid\chi^\gamma(1)$ and
$$
w_{p^k}(\lambda)=\frac{n-|\gamma|}{p^k}=a.
$$
Conversely, suppose $w_{p^k}(\lambda)=a$ and $p\nmid\chi^\gamma(1)$. Then $|\gamma|=m$ and its lower core-tower sizes are the digits of $m$. Quotient level $k$ of $\lambda$ has total size $a<p$. Each component there has size less than $p$, hence is already a $p$-core and has empty $p$-quotient. Thus $c_k(\lambda)=a$ and every higher level is empty. Combined with truncation, this gives exactly the digits of $n$, so $p\nmid\chi^\lambda(1)$.
Therefore
$$
\boxed{p\nmid\chi^\lambda(1)\ \Longleftrightarrow\ w_{p^k}(\lambda)=a\text{ and }p\nmid\chi^{C_{p^k}(\lambda)}(1).}
$$
If $k=0$, necessarily $m=0$ and $n=a<p$; the $1$-core is empty, its weight is $n=a$, and the <Hook-length formula> shows that every degree is coprime to $p$. Thus the equivalence holds in that boundary case too, with the empty partition having degree one.
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