You could use DeployAnywhere for critical and non-critical drivers. Alternatively, you could use a WMI query to select the model name, then include the variable in the path to your driver share. For this to work in your above example, the below script should return 'A120' as strName. If it returns 'Satellite Pro A120', then name your folder 'Satellite Pro A120,' because it's going to use this exact value to find the path, since we don't have conditions. Whether you use FIRM here or xcopy or something else, it should work just the same -- copy just the drivers for this model to the local drive. Then, as you mentioned, you'll reference this from your mini setup.
'On Error Resume Next
dim strName, strComputer, objWMIService, colItems, fso, objShell, objFolder, colSubfolders
strComputer = "."
Set fso = CreateObject ("Scripting.FileSystemObject")
Set objWMIService = GetObject("winmgmts:\\" & strComputer & "\root\cimv2")
Set colItems = objWMIService.ExecQuery("Select * from Win32_ComputerSystemProduct",,48)
' Get model number of system board from the Win32_ComputerSystemProduct Name via WMI
For Each objItem in colItems
strName = objItem.Name
Wscript.Echo strName
Next