We service providers like to eliminate firewalls when testing that is why the laptop connection test failing was particularly bad for Shaw. So they keep modifying that test to get one to work and wash their hands of the issue.
The other information that only rebooting the modem works. And that the issue seems to be tied to having no traffic for an extended time, that running a continuous ping fixes it. Also points to a modem issue.
As a work around to you have a monitoring system that can poll some device on the site every minute?
that might generate the needed traffic to keep the line up.
If the new static ip on the laptop works you might also try moving over to that address as an option as well.