As someone who used to manage a public school network with over 5,000 computers, I can honestly say Lansweeper is the best tool to inventory your network. And we never used lspush or lsclient.
You really need to look into active scanning. You can set it to scan AD for logon events every 15 minutes, which should be more than enough to inventory a computer before it's been turned off. And assuming students/teachers leave the computers on for more than 15 minutes, it should be enough time for your applications to get deployed.
I know technology among school districts varies widely, but you should probably make it policy that all machines stay powered on just so you don't run into situations like the one you're in. We required all machines to stay on 24/7 so they stayed up to date with MS patches, software deployments, AV scanning, etc.