When the dealer is saying that the dash needs to be removed, it usually means that they think you have an air mix actuator problem. This is also known as a ait temp actuator. I'm not convinced that the actuator is your problem though. Usually when it fails it will always blow hot (or cold) air and will be accompanied by a flashing temp display on the ECC head. If you know what to look for, you can observe the motion of the actuator through the glove box with minimal disassembly.
You first need to remove the inner glove box. There are about six 7mm screws. Once it is removed, you will see two holes in the black steel dash structure. Use a flashlight and look through the left hole. You should see something like in the second pic. the silver threaded rod is attached to the air mix actuator. This in turn moves the air mix door lever (yellowish part). With the car running, adjust the temp on the ECC head from full hot to full cold and back to hot. You should observe the threaded rod pushing and pulling the air mix door lever. It has roughly a 90* swing and should not stop or pause. If it does than the air mix door is suspect. If it moves smoothly through the entire range, it'* something else.
On very hod days, my AC blows hot ait initially. I've never timed it but it comes out not hot after maybe 10 seconds. it doesn't get really cold until I'm driving and keeping the RPMs above 1500. Then it gets really cold