UIAutomation is what I would try, however I was unable to get it to make a selection for the printer. In this case, I think SendKeys may be the easiest since it seems to be at least required to select the printer. The challenge here though is that Chrome tends to save the last printer, so you won't know what has been selected. You would need to select the first prnter in the list and then send "s" to have it select "Save as PDF" since it is the first "s".
Using UI Automation though, I can click the Save button once the "Save as PDF" option is selected with a relatively small script, but that isn't helpful unless you can get the selection made in the first place.
I will give this some thought, but if anyone else has any suggestions...
Here is a solution that can get you started...the ClickInitialSaveButton seems to have an issue, but it works sometimes. I haven't figured out why. I am also unable to get window.print to work without an exception even though it seems to work. It appears to be a threading thing because the print dialog appears but I get an exception if it isn't dismissed soon enough. I tried a few things to get it working but never got it reliably working.
The final method PopulateChromeFileDialog seems to work, but requires a few seconds after the dialog appears before it is able to locate it.
It might be a few days before I can try again to get this working, but perhaps you'll have more success and at least this gets you something to play with.