It is indeed a limitation of the Print Dialog that the number of copies must be between 1 and 999. This is a limitation "by design", and as such may not be possible to circumvent.
However, what you can try is using a JavaScript print command, using the PrintParams (which allows to cover more or less all you need). There is the numCopies property of the PrintParams Object, and the documentation does not state any maximum value. In short, the printing command would look like this:
var pp = this.getPrintParams() ;
pp.numCopies = 1234 ;
this.print(pp) ;
For all the parameters in the PrintParams Object, refer to the Acrobat JavaScript documentation, which is part of the Acrobat SDK, downloadable from the Adobe website.
Hope this can help.
Max Wyss.