Hi,
I think it would be possible, but you have to modify init() in
arduino-1.5.2\hardware\arduino\sam\variants\arduino_due_x\variant.cpp
replacing instruction WDT_Disable(WDT); by somes calls to fonctions of wdt.c in
arduino-1.5.2\hardware\arduino\sam\system\libsam\source\wdt.c
May be the most easy would be to skip instruction WDT_Disable(WDT); and configure watchdog at the beginning your setup().
According to comment at the beginning of wdt.c you have 16 seconds to perform this configuration. Would it be enough for reprogramming the Due in order to modify your sketch?
Personally, I am not the valiant enough to try it 