I think you add 1 burden. From the CRD:
If the effect of a card or special ability requires you to perform an action and you cannot, you must perform as much as you can and ignore the rest. (See limit.)
When the threats are at their max level, even then you don't get to take a wound. The Destroyer has simply no effect at that moment. It is a prevent option, option meaning you can choose to, not forced to.