The Weblink pages (TIFF) doc viewer shows a print button, which works fine. However, in our implementation of Weblink, we have had numerous issue with the TIFF pages (embedded links don't work, fillable form fields don't work, certain graphics are degraded, etc). As a workaround, we have deleted all generated pages, which forces the viewer to use the PDF viewer (for pdf documents anyway).
The problem with this PDF viewer is there's no Print button, which our customers use. The in-browser print option attempts to print the weblink frame, which is a no go. Interestingly, a CTRL+P keyboard shortcut works to print the PDF properly, as long as the document is in focus. It stands to reason that replicating that CTRL+P functionality in a print icon placed next to the download icon should be a relatively simple fix. Downloading the PDF is an option, but it is not an intuitive step to have to download a file prior to printing from a browser.