Ok, well, typically over the range microwaves are attached to the back wall using brackets that hold the microwave. It is recommended that. "THE MICROWAVE MUST BE CONNECTED TO
AT LEAST ONE WALL STUD." See pages 5 and 6 from this link:
A 1/4-inch wood panel in the back that you have now is, I'm afraid, not substantial enough to hold the microwave without it being a danger from falling or from the constant opening and shutting of the door which inevitably can be a bit rough at times.
I would think that if you could increase the thickness of the 1/4-inch panel to 1/2-inch or greater that it would be a safer installation, but not being able to see this panel and how it's supported within the existing wall, I will not be able to guarantee this will be a safe application either. Over the range microwaves are heavy and the wall studs are what holds it up within the brackets. The screws holding the brackets in place are going to show on the opposite side of the wall, which is probably unacceptable visually. So, I'm just not seeing that this option will work for your house.
Here's what I think your options will be:
1. Use a countertop microwave instead, or
2. Purchase a smaller countertop model and mount it under your wall cabinets, using an under cabinet mounting kit. Countertop models are usually not as deep (the measurement from front to back).
Here is a video showing the dismantling of a countertop microwave from the wall cabinets and the mounting of another. You can see the hardware used, as well as what I'm suggesting. https://www.youtube.com/watch?v=GjlVtRy2E0E
Here is a page of information related to #2: https://www.google.com/search?site=&source=hp&q=mounting+hardware+for+under+wall+cabinet+microwave&oq=mounting+hardware+
I think #2 is a great option and microwaves from GE, Haier (found at Lowes), among others, make smaller, compact size microwaves.
Most microwaves need a dedicated circuit for electrical pull, so be aware of this too.
I'm not sure that I gave you the answer you wanted to hear but I don't think there is any way to get around the wall and it's 1/4-inch paneling without you beefing up the wall quite a bit, which is possible but that's another subject.
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