"I've heard tell of people using curl to automate button pushing on web admin pages. Depending on the kind of button, this could be very easy. curl is a featureful commandline file-transfer tool, but it can be used to activate a web control via script, cronjob, or other task scheduler.
The tutorial Using cURL to Automate HTTP Jobs is a great resource -- section 4, Forms, may be particularly useful here.
I don't have one of these devices to play with so I can't give you specific instructions for crafting the curl command. The general idea is to
figure out what request your web browser sends when that refresh button is pushed, and get curl to send it instead.