A 'sleeping' display should not have any effect on a down/upload. Same with disk burning, AFAIK. Sleeping the screen/display is not the same as sleeping the computer/cpu.
But if you'd like another 'solution' you might take a look at <
Backdrop>. It is a simple app that will will the screen with any color you'd like (or even an image!). And, if it's the front most app, you won't see anything except the Menu Bar and the Dock (if you don't have it hidden). And with command-tab, you can always bring the download program to the front to watch, no doubt with amazement, as that process runs!
And this thing is absolutely FREE!
One of the few downloads at VersionTracker with no negative reviews!
BTW, I use Speed Download for most downloads and it can use a very, very small window with not much more than a 'barber pole' indication download progress. There's no need to use only your browser for downloading.