You should not be running delta inventories daily. Best practice is to run a delta inventory weekly, and a full inventory monthly. Daily inventories provide no appreciable benefit, but decrease performance on all workstations and the Notification Server, which must process seven times as much data as is necessary.
You can create a Targeted Agent Settings policy and apply it to laptops so that there is a Blockout period after startup. You can set the Blockout period in hh:mm, for example 00 15 for 15 minutes of blockout time after startup. You can also define a blockout time, e.g. 11a-1:30p. It's better to set it after startup and leave the other settings alone.
You can also tell it to run at the exact scheduled time, and if it isn't available, it will not run.
Connected to the network means connected to the server, yes. That's because policies are stored in an .xml file locally on the clients, so it's possible for clients to execute software even if they can't reach the NS. So a computer might download the new antivirus, leave the office, and install it as scheduled on Thursday during the maintenance window, even though they're at a hotel.
An added benefit of moving to a weekly schedule is that you can determine if it's really the inventory scan, or perhaps some other process (patching, antivirus updates, logonscripts/AD). I'd set it mid-week rather than Monday morning, since Monday morning is going to be a big updating day for laptops anyway.
Furthermore, you could choose to exclude certain laptops from inventory and just rely on basic inventory, if it really is too taxing for the old systems.
Does this help?