Nothing in life is without risk. But you can take steps to minimize that risk:
-1- Tape some foam or a rag to the mirror surface with painters tape. You may need an accomplice to change the mirror position so you can tape it on all 4 sides.
-2- Run some tape from the front of the mirror to the bottom of the housing, so the mirror will not fall in case it becomes completely detached.
-3- Have a hair dryer handy to warm up the plastic retainers behind the glass before attempting to release the mirror from the actuator.
These images below were taken on a Driver side mirror, so positions of components will generally be the reverse of yours. While these images were taken on a workbench, there is no need to remove the mirror from the vehicle for this repair.
Photo 1:
Protective tape and glass covering has been omitted for clarity.
Adjust the mirror position down and in or out as far as it will move. Choose the upper corner that protrudes furthest from the housing. In this case, I have positioned the glass down and away, so I can insert a pry tool between the upper-inner corner of the mirror and the case.
Point the hair dryer behind the glass, into the housing, so as to warm up the back side of the glass assembly. One to two minutes on medium heat should do it.
I used 2 small plastic putty knives, stacked for greater thickness and rigidity, to pry the top corner of the mirror out. Pry as high as possible on the back of the mirror assembly. Apply steady force; the plastic putty knives will flex and limit the maximum force and likelihood of damage. If the retainers have been sufficiently warmed, the mirror assembly will release at the top and swing down (there are hinges at the bottom). The mirror assembly will move out about 1/2" before the retainers release, allowing you to shift your pry tool to a more favorable position toward the top edge rather than the side.
Photo 2:
This is with the mirror swung fully open. You will have to hold it in this position (a helper comes in handy) for the next step because the hinges are under tension by design.
Photo 3:
Use a small screwdriver to push the black plastic retainer clip (red circled) away from you and up. This releases the LED cover, which will swing up and can then be removed entirely by pulling it toward you. The latch design is similar to the battery compartment on some TV remotes.
Photo 4:
Lift out the printer circuit board and release the white 2-pin connector. Use pliers to get a good grip on the connector body and pull.
When finished installing a new LED board, snap the LED cover back on (insert the end furthest from you first), and then swing the mirror assembly back up. Push it at the upper corners until the 2 retainers "click".
Test all mirror functions to ensure they work.