Does your router support other (non-HTTP) management protocols, such as SNMP or Telnet/SSH?

Most routers do. SNMP is probably the best option, if your router supports it. Net::SNMP will be of assistance with that.

But I must echo the comment made by osbosb, you should be addressing the root cause of your problem, rather than blindly re-booting when something goes wrong.

Cheers,
Darren