A connection drop like you describe is not a known issue with the 82567LM. As you already found out, at least some of what you were reading had to do with the driver used with the 82567LM-3, which is a different driver. I do not know of a specific solution, but I have some ideas that might help track down the source of the issue. You may have already thought of some of these things.You already tried multiple driver versions, so I would look at other possible causes.
Are you using a prepared image or installing the 32-bit Windows 7 from install media? If you have a prepared image, you might try a fresh install to see if that makes any difference just in case some Windows file or registry setting is causing an issue.
Do you have Service Pack 1 installed or not? Does SP1 have any affect on the issue?
Try running the diagnostics available from the Link Speed tab in Windows device manager. Is there any difference in the results of the diagnostics when the network connection is acting normally versus when no traffic is flowing? Are there any indications in the diagnostics that anything is failing?
How are your speed and duplex settings configured? Auto? What about on the link partner? Does forcing the speed and duplex to a different setting on both ends of the connection cause the connection drops to disappear?
Are you running the latest BIOS version on the laptops? Does upgrading the BIOS make any difference?
Have you tried running without any security software (e.g. antivirus, firewall, intrusion detection) installed?
Let me know what you find.