This is a fascinating question, and I freely admit that I have absolutely not the slightest idea of the answer, but I recommend sending an email to Hpa.lincolnquestions@illinois.gov which should reach the right people at the Abraham Lincoln Presidential Library.

Who's Willie?

I can't speak to letters sent as condolences, but I can tell you that his death was not widely reported on in the South in newspapers, and mostly in the Upper South. The few articles I have found were of a conciliatory tone but usually surrounded by other news that mocked the Lincolns.