To fix this problem, I installed the Greasemonkey firefox addon and then created this simple Greasemonkey script to trick Google into "ungreying" those download links.
Now I can click "Download to Picasa" and have it download all of those photos. My first trial download hasn't finished yet because it doesn't seem to care that 272 photos have already been downloaded, and so I'm not quite sure what the ultimate outcome will be... However, I think this is progress.
NOTE: For Linux users, adding a single line to the file ~/.mailcap that has in it:application/x-picasa-detect; false; description=Picasa Installation detection(in all one line; no wrapping) will also ungrey "Download to Picasa". Unfortunately, that only seems to convince PicasaWeb that Picasa2 is installed. To get all four options ungreyed, the force script is needed.