Again, I would recommend turning Energy Saver "Off" to be 100% sure your Mac going to sleep at 2am is the problem… it shouldn't be, but just to be sure.
Certainly, your transcript shows repeated dropped connections throughout the multi-file, multi-connection, transfer (even at 2am for example). In every single case Yummy reconnects, asks the server how much of the file its got, then resumes uploading at that exact place, so I can't see that Yummy is doing anything wrong at all. All other FTP apps would have given up long ago

I do note that the server is PureFTPd and this keeps temporary files while you're uploading, so maybe it's got some bug with resuming reliably.
Perhaps you have a problem with your Internet connection. I note from the transcript that the server reports quite slow speeds for completed transfers, and I also notice many short periods (e.g. 30 - 60 secs) where there is simply no response from the server at all to the most simple of commands prior to a transfer.
So I would recommend checking with the server provider to check that there are no issues there, and also have your own Internet connection checked for reliability.