I cut your script down to just the basics and it works for me. I did have to change the connection command to work in my environment but other than that and registering a default instance of SQL Server there were no errors. Once I did a refresh of the CMS server the newly registered server was visible and accessible.
[System.Reflection.Assembly]::LoadWithPartialName('Microsoft.SqlServer.SMO') | Out-Null
[System.Reflection.Assembly]::LoadWithPartialName('Microsoft.SqlServer.Management.RegisteredServers') | Out-Null
$CMSInstance = 'CMS_ServerName'
$connectionString = "Data Source=$CMSInstance;Initial Catalog=master;Integrated Security=SSPI;"
$sqlConnection = new-object System.Data.SqlClient.SqlConnection($connectionString)
$conn = New-Object System.Data.SqlClient.SqlConnection("Server=$CMSInstance;Database=master;Integrated Security=True")
$CMSStore = new-object Microsoft.SqlServer.Management.RegisteredServers.RegisteredServersStore($conn)
$CMSDBStore = $CMSStore.ServerGroups["DatabaseEngineServerGroup"]
$RegServerName = 'ServerToRegister'
$RegServerInstance = $RegServerName
$NewServer = New-Object Microsoft.SqlServer.Management.RegisteredServers.RegisteredServer($CMSDBStore, "$RegServerName")
$NewServer.SecureConnectionString = "server=$RegServerInstance;integrated security=true"
$NewServer.ConnectionString = "server=$RegServerInstance;integrated security=true"
$NewServer.ServerName = "$RegServerInstance"
$NewServer.Create()