Thought ..... the 2 wires from the drive motor of the doors reconnect to the servo board where the motor connections were made and use the electronic circuitry as a speed controller - adjusting the variable resistance pot to the correct speed of the door opening and closing - then with a bit of superglue lock down the variable resistor so it wont move - hey presto - doors slowed down and operating. If you don't have the innards, I suppose you could strip down a redundant servo and add that ..... just another thought. Think about these guys who do plastic magic and use servos to drive the models.
When using the servo board as an ESC, I always used the transmitters trim slider as the speed control to make it proportional control rather than full on, maybe off.
Fixing the pot when using the servo board as a speed control only sets the center off when compared to the signal coming in to it.From that, it very rapidly transitions to full speed from a virtually zero deadband. That's why I suggested the very small 10A ESC from China. Nice big deadband, more gradual control, not much bigger than a standard servo board. There are smaller ones, but they don't come with wiring attached so you need to solder your chosen wiring to them.
The alternative of a servo stretcher/rate tweaker should also be fine. Either way, if it is part of a sequence being controlled by a single button press, it will need a source of signal to send it from open to closed, and, presumably, back again at the right time.
If not relying on eyesight to stop it in the right place, some sort of limit switch will be needed to either cut the servo motor supply or tell the control box to stop.