Heat from drilling, sanding, and friction finishes can lead to cracking. You might not even notice it until after hours or days later. So in my opinion, friction polish is out.
I would recommend applying shellac after 400, then for a finer finish, sand through with 600 - 800 grit and finish your trumpet with MinWax PolyCrylic gloss. About 4 coats. They have it at Walmart and the hardware stores.
With oily woods, I usually remove the surface oil with DNA, and apply shellac as a sealer. Just about any kind of finish can be applied on top of the shellac after it is dry.