Need help writing the contents of System.Data.DataSet to a SQL server table

Please help with this simple script. I am trying to get a list of all user databases from a server, then write that list to a central table. So far I am able to query a remote SQL instance, and display the results on the screen. However I am struggling with how to write the values in System.Data.DataSet to a SQL Server table.

The table actually has 5 columns and allows null values. I only need to insert INSTANCE NAME and DATABASE NAME to the table.

Everything works up to line 19, but I am stuck on how to insert the data to a table.

# create the connection to the remote instance
$sqlConn = New-Object System.Data.SqlClient.SqlConnection
$sqlConn.ConnectionString = "Server=my_server;Integrated Security=true;Initial Catalog=my_db"
$sqlConn.Open()

#build cmd sting and run the command
$sqlcmd = New-Object System.Data.SqlClient.SqlCommand
$sqlcmd.Connection = $sqlConn
$query = "SELECT @@SERVERNAME as SQL_INST_NAME, [name] FROM sys.databases WHERE database_id > 4 AND [name] <> 'someOtherDB' "
$sqlcmd.CommandText = $query

#bind results to data adapter
$adp_rs = New-Object System.Data.SqlClient.SqlDataAdapter $sqlcmd
$data = New-Object System.Data.DataSet
$adp_rs.Fill($data) | Out-DataTable

#display data: this shows me what I expect to see
$data.Tables

#connect to other SQL instnace
$localConn = New-Object System.Data.SqlClient.SqlConnection
$localConn.ConnectionString = "Server=MyServer\MyInstance;Integrated Security=true;Initial Catalog=db"
$localConn.Open()

$bc = New-Object ("System.Data.SqlClient.SqlBulkCopy") $localConn
$bc.DestinationTableName = "dbo.table_name"
$bc.WriteToServer($adp_rs)
$localConn.Close()

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