From within a PDF I don't see a way to realize this.
With JavaScript and the PrintParams properties you can handle a lot of things, but you don't can select a specific paper tray, cause the neccessary parameters vary from brand to brand, from driver to driver and from emulation to emulation.
That's why people often have strange effects when printing Office documents.
There you can set a specific paper tray for the documents, but when they use other printers than the author it goes bedlam.
The only thing how I can image is to install a second printer driver "Lexmark X782 Manual Tray" with the manual tray set as default tray.
Then you can use the PrintParams to direct print out to this specific printer.
var pp = event.target.getPrintParams();
pp.printerName = "Lexmark X782 Manual Tray";
event.target.print(pp);