Unfortunately, I don't think that's possible, since badges are rendered using code that needs to be included in the app: Finder would need to integrate the code to tell its badge to change.
Consequently, third party apps, in theory, could get the file count of your Downloads folder, and they could code a badge to display it, but they wouldn't be able to access the Finder's dock tile objects: each app can only access their own, and app icons can't only be placed on the area of the dock reserved to folders and document aliases.
I couldn't find any utility that could solve this issue. I'm sorry. It's a great idea though, and I would totally love it if someone proved me wrong and posted an app capable of doing this, for I too have a pretty messy downloads folder.
The only way I found to get easily the item count is the status bar. Perhaps though, you could do a workflow in Automator or Apple Script Editor. I know Automator can make folder actions, which are run every time a file is added to a folder. It would just be a matter of keeping a count, I guess.
Sources: A bit of research on desktop badges in Apple's developer documentation. The examples there all show the desktop tile object needs to be accessed directly within code.
Hope it helps! And good luck.