Is there a PuTTY variant without modal message boxes?
An alternative using some simple AutoHotKey scripting:
WaitForError:
WinWait, PuTTY Fatal Error
WinClose
Goto, WaitForError
Save the above as file.ahk and run it (after installing AutoHotKey on your PC).
This script automatically closes any dialog boxes titled "PuTTY Fatal Error". It waits persistently in the tray area.