The state of the monitor isn't relevant unless an interactive user does something on the physical machine. If there is no active user on the physical machine then the monitor status is irrelevant. For example if a user remotes into the machine then the local monitor won't kick on.
You didn't specify how you schedule this morning work but I assume you'd use Task Scheduler. Therefore TS wouldn't trigger any monitor either as there is no physical user logging in. Of course booting up the computer would be a physical action and would trigger the monitor but I'd have to ask why you are rebooting the machine at all. Just let it run all the time and schedule the work using TS.
The only way I could possibly see this working is if you set up a scheduled task that ran every time windows starts that turns off the monitor. But I don't agree with that solution as it would mean a user who walks up to the machine and turns it on will see the login screen and then at some point thereafter it would shut down, potentially while they are using it. They would then need to turn it back on.