You are correct, there is no native way to do a ctrl-v from within the command prompt window. However using autohotkey you can enable a hotkey to do the same function. After installing autohotkey, add a few new lines (provided below) to your existing script and you will be ctrl-v, ctrl-v ing away.
#IfWinActive ahk_class ConsoleWindowClass ^V:: SendInput {Raw}%clipboard% return #IfWinActive