My Chase Sapphire Preferred Visa benefits guide says absolutely nothing about discounts or coupons voiding the insurance. It says you must complete the entire rental transaction using your card. So if the rental doesn't cost you anything because you're using free days, then it wouldn't apply. But if you rent for 8 days and use one free day to get the weekly rate, you'll be fine. I've used a free day on a 4 day rental and was able to claim an accident via Chase no problem.
Initiate and complete the entire rental transaction using your card that is eligible for the benefit.
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I wouldn't use a corporate code that includes CDW, as that would waive the insurance per the benefit guide details. Coupon through National, no problem.
All that said, your best bet is to find the benefits guide for that particular card, and read it carefully. Then call if you have questions. I wouldn't rely on someone posting on this forum for a definitive answer, as it won't hold any water when trying to file a claim.