VMware Support came up with a solution.
Big thanks to Jain 
The problem was basically how to "not set" the name parameter from within PowerShell.
This is a version of the script that forces VI to use the DisplayName when registering a .vmx file.
$folder = Get-View (Get-Datacenter -Name <datacenter> | Get-Folder -Name "vm").ID
$pool = Get-View (Get-Cluster -Name <cluster> | Get-ResourcePool -Name "Resources").ID
$esxImpl = Get-VMHost -Name <ESX-host>
$esx = Get-View $esxImpl.ID
$dsBrowser = Get-View $esx.DatastoreBrowser
foreach($dsImpl in $dsBrowser.Datastore){
$ds = Get-View $dsImpl
$vms = @()
foreach($vmImpl in $ds.Vm){
$vm = Get-View $vmImpl
$vms += $vm.Config.Files.VmPathName
}
$datastorepath = ""
$searchspec = New-Object VMware.Vim.HostDatastoreBrowserSearchSpec
$searchSpec.matchpattern = "*.vmx"
$taskMoRef = $dsBrowser.SearchDatastoreSubFolders_Task($datastorePath, $searchSpec)
$task = Get-View $taskMoRef
while ($task.Info.State -eq "running"){$task = Get-View $taskMoRef}
foreach ($file in $task.info.Result){
$found = $FALSE
foreach($vmx in $vms){
if(($file.FolderPath + $file.File[0].Path) -eq $vmx){
$found = $TRUE
}
}
if (-not $found -and $task.Info.Result[0].File -ne $null){
$vmx = $file.FolderPath + $file.File[0].Path
$params = @($vmx,$null,$FALSE,$pool.MoRef,$null)
$folder.GetType().GetMethod("RegisterVM_Task").Invoke($folder, $params)
}
}
}