We have this type of discussion periodically here. Rates are quoted based on the forecast of future supply and demand conditions at the time of reservation. Changing the reservation at a later time usually means those original utilization conditions have changed, thereby resulting in a higher rate.
Rental agencies do not want customers to lock in a low rate far in advance and then add more days for the same price at the last minute. If they did, everyone could theoretically book the car way in advance at a low price for every day in the peak season, and then just change and cancel at the last minute what they don't need. This would create a lot of uncertainty in terms of how many cars the rental agency will actually rent, and thus how many they need. Worst of all is the opportunity cost: The car could have gone to another customer who made the reservation on shorter notice and is willing to pay much more.
You are right in that trying to get more money from you specifically will backfire if you go to a competitor. However, as explained above, the car will likely go to more profitable, last minute customer(s). Several of them on shorter rentals, but at a higher price per day, will probably be more lucrative than giving the car to you for 12 days, even accounting for more staffing and cleaning costs with a higher transaction volume.