[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.
| $n$ | $|[Q]_\mathrm{res}|$ |
|---|---|
| $3$ | $1$ |
| $4$ | $2$ |
| $5$ | $3$ |
| $6$ | $5$ |
| $7$ | $5$ |
| $8$ | $8$ |
| $9$ | $10$ |
| $10$ | $15$ |
| $11$ | $19$ |
| $12$ | $31$ |
| $13$ | $41$ |
| $14$ | $64$ |
| $15$ | $94$ |
| $16$ | $143$ |
| $17$ | $211$ |
| $18$ | $329$ |
| $19$ | $493$ |
| $20$ | $766$ |
| $21$ | $1170$ |
| $22$ | $1811$ |
| $23$ | $2787$ |
| $24$ | $4341$ |
| $25$ | $6713$ |
| $26$ | $10462$ |
| $27$ | $16274$ |
| $28$ | $25415$ |
[/greyproof]