If you use a corporate code you aren't entitled to you could be in trouble... you might (or might not, most likely not - especially if you're a #1 gold member) be asked for id when you pickup the car. The CDP you use is available to the people doing your reservation, and will be on the reservation slip you're handed

However, if you have an accident this is when the trouble will (?may?) strike - in this scenario Hertz will, most likely, be in contact with the company of the CDP - especially if they need insurance details (as with some companies that self-insure) - for example my corporation has a "great" CDP because it self-insures (so really reduces the price). If this does happen then they'd find out you weren't entitled to the CDP and its likely you would be liable for the complete damage from the accident, plus a risk of being charged with driving without insurance. As such, I'd check at a minimum whether the CDP you are using is so good because it comes with no real insurance.

Having said this, the above is the worst case scenario, and may be scaremongering, but, its a risk you have to accept. If you get 'caught' on trying to get the car they'll probably bump you up to rack rate and it might be more expensive than if you used a CDP that is publically available. Its a bit different than using corporate codes on hotels due to the accident risk / legal liability etc parts.

Having said that, its your choice - with regards to the coupon... the language there seems to imply that the PC you've put in does have some form of coupon to be eligible? Difficult to say without knowing the PC, but, sometimes they do expect something to hand in (such as an internet printout / eligibility) - or they might just ask you casually where you saw the coupon (so make sure you can answer correctly on that front).