Man, I love this idea. I wish I had seen it years ago... Before I had the lvp flooring.
For an adhesive, i think anything for vinyl planks would be fine. This is a small area and the concern over expansion/contraction isn't as prevalent as it would be in an entire room or house. Just hit up the local hardware store and see what they've got.
That being said, I'm concerned about the thickness of your base material. If there's no support underneath the osb or ply, there's a really good chance it will start sagging and breaking.
Also, a lot of vinyl plank flooring like that is intended to float, so trim like the stuff in the example pic would be best. Plus adding trim will help keep the whole thing from separating/splitting on the edges as well as adding a nice finished look.
All that to say, it's a great little project and I'm eager to see the final product!