[greyproof=RMCS vs. $n$] Seem to have $[Q]_\mathrm{res}$ that grows exponentially in cardinality in $n$. Here is sage code that plots these quivers:
Code used to find the restricted mutation classes (requires this version of sage):
Here is a table consisting of $n$ and the corresponding $[Q]_\mathrm{res}$ cardinalities.
| $n$ | $|[Q]_\mathrm{res}|$ |
|---|---|
| $5$ | $6$ |
| $6$ | $2$ |
| $7$ | $28$ |
| $8$ | $42$ |
| $9$ | $48$ |
| $10$ | $440$ |
| $11$ | $4000$ |
| $12$ | $768$ |
| $13$ | $50585$ |
| $14$ | $62630$ |
| $15$ | $86369$ |
[/greyproof]