My router got updated and my initial experiments had all been on a tmpfs. As a reminder to my future self, JFFS2 should be enabled for persistent storage on flash (or use a USB drive of some kind).
Following https://www.dd-wrt.com/wiki/index.php/Journalling_Flash_File_System#Directions_for_.28normal.29_users:_using_Web-GUI_Interface worked for me.
I then created a script on the jffs2 mount point and updated DNSMasq settings (specifically "Additional DNSMasq Options") to point to this, e.g.:
Where /jffs/dhcp_ha_post is executable:
chmod a+rwx /jffs/dhcp_ha_post