How do I write to csv file, naming the file using the value of the first row?

Starting with your secondary question. With the dates formatted how you have them, a simple solution could be to sort numeric descending. A more robust solution would be to use a DateTimeStamper to create a new attribute for the current date, and then use a DateTimeCalculator to determine the interval between the input dates and the current date to find the most recent.

For your primary question, there are probably a lot of ways you could do this. One way would be to use a Sampler on your now sorted dates to get the feature with the most recent date. An attribute manager can be used to rename the CompleteDate to something like MostRecent, and remove all other attributes. Then you can use a featuremerger with a join on of 1 and 1, to append that attribute to all features (including the one that was sampled). When you go to write out your data you can fanout on that attribute, but not write it to the csv.

I know that the AttributeManager is the preferred answer, but in this scenario I find it easier to configure using a VariableSetter and VariableRetriever.