First of all welcome to FlyerTalk! What contract ID are you using for the Alamo rate? I agree that the Hertz one is risky; they'd generally ask for ID on that one. Thrifty and Dollar make it a policy of not asking for ID on corp accounts except for Government or staff, and National seems to be the same. It's going to be a lot cheaper picking up on the 10th since you won't be paying airport tax or your hotel's parking fee. The other thing to consider is luggage: If you get something like the Sebring, where the soft top folds into the trunk, there's room for carry-on at best. This happened to me in Toronto and I wound up returning the convertible a day early and picking up a Cobalt just for the last day.
Also be careful about making back-up plans on a convertible. Most agencies consider it a guaranteed vehicle as they are in high demand and often short supply at a sun destination. Failing to cancel without adequate notice (usually 24 hours) could leave you paying one day's charge as a penalty, and likely at the rack rate.
Let us know your parameters and we'll help you find the best deal using a safe CDP/contract ID. I don't condone using any contract IDs you don't qualify for, but qualification is a loose term in and of itself. Basically, use the same degree of honesty that you would when completing your tax return.
Happy travels!