Here's the answer.
A Young diagram (YD) with $k \le N$ boxes is a partition of $k$, $[\lambda_1... \lambda_k | \lambda_1 \ge \lambda_2 ... \ge \lambda_k]$; these are in 1-1 correspondence with the irreps we find in $\mathcal{H}_{fund}^{\otimes k}$. We denote it by $[\lambda]$; it turns up in $\mathcal{H}_{fund}^{\otimes k}$ $\alpha_{[\lambda]}$ times.
Let's use this short note as a reference on how to calculate the dimension of the irrep corresponding to a Young diagram.
At leading order in large $N$, the dimension of $[\lambda]$ is
$$\text{dim} \mathcal{H}_{[\lambda]} = \frac{N^k}{hl([\lambda])},$$
where $hl([\lambda])$ is the hook length of the YD. Here, we have approximated the numerator by $N^k$ but left the denominator exact; this will be justified a posteriori. What we want to prove, then, is
$$\alpha_{[\lambda]} = \frac{k!}{hl([\lambda])}.$$
Notice that this is trivially true for $k=1$. We prove it for generl $k$ by induction.
From a YD with $k$ boxes, we can make one with $k-1$ boxes by removing a "corner" box, which corresponds to reducing one of the $\lambda_i$ by $1$. Denote such a young diagram with $[\lambda]\setminus i$.
(If $\lambda_i = 0$ then $[\lambda] \setminus i$ doesn't exist. We assume that these are removed in the discussion below.)
$[\lambda] \setminus i$ appears in $\mathcal{H}_{fund}^{\otimes k-1}$ $\alpha_{[\lambda] \setminus i}$ times. Since $\mathcal{H}_{fund}^{\otimes k} = \mathcal{H}_{fund}^{\otimes k-1} \otimes \mathcal{H}_{fund}$, all the ways to arrive at the irrep $[\lambda]$ correspond to adding a box in the $i^{th}$ row to $[\lambda]\setminus i$, giving
$$\alpha_{[\lambda]} = \sum_{i=1}^{k} \alpha_{[\lambda] \setminus i} \quad \Rightarrow \quad \sum_i \frac{\alpha_{[\lambda]\setminus i}}{\alpha_{[\lambda]}} = 1.$$
It turns out that this is a rather involved statement to prove, but it is one of the fundamental facts about the hook length, and the proof, via a "Hook walk" is sketched on wikipedia (Original paper by Greene, Nijenhuis, Wilf). $e_{\lambda}$ on wikipedia and $F$ in the paper is our $\alpha_{[\lambda]}$.
I thank Cosmas Zachos for their comment, which helped me put the pieces together (esp. for making me realise that large $k$ isn't necessary).
An interesting side-note: this formula relates the dimension of the irrep to the number of standard Young tableaux (sYTs), whereas states are indexed by semi-standard Young tableaux. The dimension can be written as ${N \choose k} (\# \text{sYTs with $k$ boxes})$, i.e. we pick $k$ numbers from $1... N$ and then use these as $k$ labels for making sYTs.
