I went and tried adding Installdate and Lastchanged from tblSoftware but that didn't seem give me correct values. Then I tried tblSoftwareHist Installdate and Lastchanged which may have given correct values. (I don't know the difference between the two tables.)
tblSoftware is the list of currently-installed software.
tblSoftwareHist, as the name suggests, is a history of changes: software installed and removed since the asset was initially scanned.
According to the database documentation, 
InstallDate is the date the software was installed and 
LastChanged is the date the data table was last updated. 
LastChanged won't likely be be exactly the date the software was installed, but it shouldn't be more than a couple of days out, depending on how frequently your LANSweeper scans for software changes. Those values from 
tblSoftware should reflect when the currently-installed version was installed and the table entry updated.
From my inventory, it's not uncommon to have no 
InstallDate value recorded, so I'll use 
IsNull(InstallDate, LastChanged) below to fall back on 
LastChanged where there is no 
InstallDate value.
Select Top 1000000
  tblassets.AssetID,
  tblassets.AssetName,
  tblassets.Username,
  tsysAssetTypes.AssetTypeIcon10 As icon,
  tblassets.IPAddress,
  tblassets.Lastseen,
  tblSoftwareUni.softwareName As Software,
  tblSoftware.softwareVersion As Version,
  tblSoftwareUni.SoftwarePublisher As Publisher,
  tblSoftware.InstallDate,
  tblSoftware.LastChanged,
  ( SELECT                                                  -- if InstallDate is null, use LastChanged as a better-than-nothing alternative
      Max(IsNull(tblSoftwareHist.InstallDate, tblSoftwareHist.LastChanged))
    FROM
      tblSoftwareHist
    WHERE
      tblSoftwareHist.AssetID = tblAssets.AssetID           -- for this computer
      AND tblSoftwareHist.SoftID = tblSoftware.SoftID       -- and this piece of software
      AND tblSoftwareHist.Action = 1                        -- installed
                                                            -- prior to the current install
      AND IsNull(tblSoftwareHist.InstallDate, tblSoftwareHist.LastChanged) < IsNull(tblSoftware.InstallDate, tblSoftware.LastChanged)
  ) AS PreviousInstall,
  tsysIPLocations.IPLocation,
  tblAssetCustom.Manufacturer,
  tblAssetCustom.Model,
  tsysOS.OSname As OS,
  tblassets.Lasttried,
  tblSoftware.Lastchanged
From
  tblassets
  Inner Join tblAssetCustom On tblassets.AssetID = tblAssetCustom.AssetID
  Inner Join tsysAssetTypes On tsysAssetTypes.AssetType = tblassets.Assettype
  Inner Join tsysIPLocations On tsysIPLocations.LocationID = tblassets.LocationID
  Inner Join tblState On tblState.State = tblAssetCustom.State
  Inner Join tblSoftware On tblassets.AssetID = tblSoftware.AssetID
  Inner Join tblSoftwareUni On tblSoftwareUni.SoftID = tblSoftware.softID
  Left Join tsysOS On tsysOS.OScode = tblassets.OScode
Where
  tblSoftwareUni.softwareName Like '7-Zip%'
  And tblSoftware.softwareVersion Not Like '19.00%'
  And tblState.Statename = 'Active'
Order By
  Version,
  tblassets.Lastseen Desc,
  tblassets.AssetName