Instead of directly printing from the browser, you could have your application produce PDF files - you can then make used of a 'watched folders' function, which prints any file that lands in the folder to a specific printer. Could be a bit tricky to set up, but seeing as it's an internal app, it may be possible. It's the only way I can think of to completely automate it cross-browser.
The flow would go something like:
- User requests the printable page and label page from the app
- App produces two PDFs and saves them to specific, separate folders
- Another program sees the new files, and automatically prints them to the right printer.
It could be tricky, depending on how many instances of the printers there are, because you'll need to save them at the server side, rather than anywhere on the client, to avoid the user being asked about where to save the file. If there's a location on the network available to both the application server and the printer, this might not be such a big deal.
There are plenty of server components to produce PDF files, and desktop apps to watch folders for incoming files to print them. Which of these suits you (if any), will depend on your environment. Something like this: http://www.pdfstore.com/details.asp?ProdID=703 might work for the printing side of things...