I have rented from companies at the airport and off-airport, and always go for the cheapest. Depending on how far away from Ponta Delgada you are staying, you can still rent a small inexpensive car and have a taxi bring your luggage to where you are staying and then get a taxi when you leave back to the airport. Just do the math to see if this method will save you money. I can give you a ballpark figure for a taxi if you let me know where you are staying.
Also, I never lock in a price with any company. I ALWAYS make a reservation that can be changed or cancelled in case I see a better price. You can still pre-pay for a car reservation, just make sure in the fine print that you can cancel with no fees. I have used “Autoslash.com” and saved many times with their free service.
Have you already received my PDF guide to the Azores? Possibly another Susan sent me a request, but I can send you the link to it if you want. My email is: email@example.com. In the subject line, type: Azores.