Well just like with car paint least aggressive first because again you are removing material, there are no two ways about it.
For those kind of scratches I would tape off the engravings and start with a dremel and 520 polishing bit. It has silica carbide and a lubricant already built in so it's pretty hassle free to use. Go slowly and it should take them right out. Then finish the whole thing off with flitz or cape cod cloths.
A buffing wheel and compound will certainly get the job done very well and quickly but there isn't enough precision to avoid the engravings. If your father in law has a lot of experience with it he will be able to tell you about the engravings. It's hard to make a 100% recommendation without actually feeling the scratches and checking the depth of the engravings.
Ill write this because nothing is easy: If you have a dremel and pick up a 4.00 520 bit for it do yourself a favor and practice on the back of a scratched up SS spoon from your cutlery drawer. Being that the tube is contoured you run the risk of letting the dremel get away from you and you REALLY don't want the fast spinning metal collar contacting your
device. The bit itself can't damage your item but the metal collar of the dremel can really mess it up. If you aren't comfortable with it don't do it. The back of the spoon will at least give you an idea how it can pull when going down a rounded surface.