Removing the door panel was easy -- remove 6 Torx screws, and the panel falls off. Next you need to remove an electrical plug and 2 individual wires from the old dispenser. The plug came right off, but the wires were not easy to access, and they fought me more than I expected. With some pliers and wiggling, they eventually let go and came free.
The dispenser is held in place with 6 Phillips screws through 2 removable metal brackets, and those came out easily. The toughest part of the job came next. The metal around the hole in the door in which the dispenser sits is folded forward, creating a sort of elongated spring clip that latches against 6 plastic tabs on the bottom side of the dispenser, 3 above and 3 below. This metal had to be carefully pried back away from the plastic tabs, freeing one tab at a time, but I had to keep applying inward pressure on the dispenser to prevent it from being pulled back into place. Once freed from this metal rim, the dispenser came free.
When reassembling, getting that metal rim to seat against the plastic tabs was again a problem. I replaced the metal brackets and Phillips screws but did not tighten the screws completely. This compressed the gasket around the dispenser and got the new part almost into place. I was then able to work the metal rim onto one plastic tab at a time. Once all 6 tabs were engaged, I tightened the screws completely. The rest was easy -- replace the wires and plug, replace the door panel, and turn the power back on.
My advice to you is to expect delays when dealing with that folded metal rim, both disassembling and reassembling. Be patient and let the screws do the work of applying pressure for you so that you don't break your brand new replacement part.
For the record, replacing the dispenser solved my problem. The soap dispenser now opens at the appropriate time in the cycle.