Yes. If you were to use a stud finder, you'll be able to tell exactly how wide that board is behind the paneling. You may be able to simply press and feel the movement of the paneling and you can tell where it's reinforced. Or, if you have the ear for it, simply tap with your knuckles and you can hear the difference in sound and tell exactly where the edges are.
$165 is ludicrous for something so easy to do yourself.
May I give you a tip, use a mount where you can remove the television from the wall when you are on the road. Although there is a place for mounting there, the television will no doubt still bounce around, weakening the screws over time. If you can remove the television from the mount and simply lay it on the bed or a couch, it will save the television and save the wall. It will also make it possible to take the television outside if you want,or bring it in the house if you have to store the camper at a facility and you are concerned about theft. Lots of reason for an easy hook, type mount.
I don't have good photos of my current camper, but this is what I'm talking about in these photos from my previous camper. When we sold the camper, I removed the wall mounts and put them in the new camper. Very easy to just lift off, and saved a world of concern for possible damage to the television or the wall:
I searched through my photos and did find this one. This is a photo of the same mount installed on our current camper. I did not have the television hanging here when the photo was taken. Like yours, there is a red and white label on the wall stating to mount the television "here". If you can blow up or zoom up the photo, you'll see the label under the mount.
Four screws and this is attached to the wall. No need to pay anyone else to do this:
