I really went back and forth about installing a TV in the kids bunk. My opinion is they should be out in the woods enjoying the outdoors. But on days with bad weather, It's a great sitter.
You don't have a lot of room on the backer, so you need to be careful. The "Backer" is about 5 inches to each side from center and about 6 inches top and bottom from center.
I installed a small TV and used 1 1/4 inch screws. Unless you are installing a 55 inch flat panel, that should suffice.
I would use the smallest mount you can find. I used a $19.00 mount I was able to find at Micro Center. I'm sure amazon has a small one as well.
Picture of Mount
And TV. This bunk is used by my son and daughter, so this size is just perfect for them.
Hope that helps, but I'm sure others will have additional info to add and always welcomed.
Remember, if you go thru the wall, at least you'll have a place to hang your towel in the center of the shower.