I did exactly the same thing, and securing the new microwave was pretty easy. It has worked great for several years and over 10,000 miles on some occasionally horrible roads. It's very solid and secure.
I removed the old microwave and bracket, and built a wide shelf for the bottom of the opening out of some clear fir boards I had in the shop because when polyurethaned, it was very similar in color to our cabinets. I did have to dowel and glue two boards to get the width I needed, but two separate boards fastened in place would have worked too. After marking where the microwave feet would sit, I make four wood donuts with a circle cutter, and glued them to the shelf so the feet would sit inside. I could have just used a spade bit to drill some recessed holes too. Then, I went to a local Ace Hardware to their rope section where they sold nylon strapping by the foot along with various fittings. Picked up enough strap to go over the microwave, and a buckle too. Cut the strap in half, added the buckle, and cut to length with enough extra to go slightly under the sides of the microwave and secured the ends with fender washers and screws. You can't see the strap without looking for it because the black strap matches the black microwave sides.
To cover all the holes on the back wall, I used 1/4" plywood and covered it in fabric to match our Safari themed interior and filled in the the recess inside the top of the opening with plywood. On the right side where there was pocket of sorts, I added some wood shelves with thin fronts to make the area a spice rack. It's worked great. I also ran some wires up from the under cabinet light and added a small LED automotive accent light. I chose a smaller microwave which also left room on the right for placing stuff while parked.
One thing I decided to do for safety, because the microwave is above our propane stove, was to make a heat shield for the bottom of the shelf. Found a precut piece of sheet metal at Home Depot that was the right size, along with 6 1/4" nylon spacers and screwed this to the bottom of the shelf after spraying the sheet metal with flat black heat resistant BBQ paint. The sheet metal extend just about 1/4" out from the front edge of the shelf.
First photo shows how I removed the very heavy convection microwave, the second, the opening after the bracket was removed, and the third, the finished project.
Tom and Pris M. along with Buddy the 17 year old Siamese cat
1998 Safari Serengeti 3706, 300HP Cat 3126 Allison 3060, 900 watts of Solar.
Dragging four telescopes around the US in search of dark skies.