service restart_dnsmasq is the correct command.
Thanks! Here's the log after the correct command:
May 16 13:27:47 rc_service: service 4189:notify_rc restart_dnsmasq
May 16 13:27:48 dnsmasq[462]: exiting on receipt of SIGTERM
May 16 13:27:48 custom config: Appending content of /jffs/configs/dnsmasq.conf.add.
May 16 13:27:48 dnsmasq[4194]: started, version 2.73rc1 cachesize 1500
May 16 13:27:48 dnsmasq[4194]: warning: interface ppp1* does not currently exist
May 16 13:27:48 dnsmasq[4194]: asynchronous logging enabled, queue limit is 5 messages
May 16 13:27:48 dnsmasq-dhcp[4194]: DHCP, IP range 192.168.11.100 -- 192.168.11.254, lease time 1d
May 16 13:27:48 dnsmasq[4194]: using nameserver 64.145.73.2#53 for domain bbc.co.uk
May 16 13:27:48 dnsmasq[4194]: using nameserver 209.107.219.3#53 for domain bbc.co.uk
May 16 13:27:48 dnsmasq[4194]: using nameserver 64.145.73.2#53 for domain netflix.com
May 16 13:27:48 dnsmasq[4194]: using nameserver 209.107.219.3#53 for domain netflix.com
May 16 13:27:48 dnsmasq[4194]: using nameserver 64.145.73.2#53 for domain overplay.net
May 16 13:27:48 dnsmasq[4194]: using nameserver 209.107.219.3#53 for domain overplay.net
May 16 13:27:48 dnsmasq[4194]: read /etc/hosts - 5 addresses
May 16 13:27:48 dnsmasq[4194]: read /etc/hosts.dnsmasq - 12 addresses
May 16 13:27:48 dnsmasq-dhcp[4194]: read /etc/ethers - 12 addresses
May 16 13:27:48 dnsmasq[4194]: using nameserver 62.241.198.245#53 for domain local
May 16 13:27:48 dnsmasq[4194]: using nameserver 62.241.198.246#53 for domain local
May 16 13:27:48 dnsmasq[4194]: using nameserver 64.145.73.2#53 for domain bbc.co.uk
May 16 13:27:48 dnsmasq[4194]: using nameserver 209.107.219.3#53 for domain bbc.co.uk
May 16 13:27:48 dnsmasq[4194]: using nameserver 64.145.73.2#53 for domain netflix.com
May 16 13:27:48 dnsmasq[4194]: using nameserver 209.107.219.3#53 for domain netflix.com
May 16 13:27:48 dnsmasq[4194]: using nameserver 64.145.73.2#53 for domain overplay.net
May 16 13:27:48 dnsmasq[4194]: using nameserver 209.107.219.3#53 for domain overplay.net
May 16 13:27:48 dnsmasq[4194]: using nameserver 62.241.198.246#53
May 16 13:27:48 dnsmasq[4194]: using nameserver 62.241.198.245#53