I figured giving myself a consistent "ground" to aim at with the router was probably them easiest way to go. It will also make it easier to balance up the two sides.
I'll clean the surface up with a gouge so the machining marks will disappear. You know how closely it will be examined once I'm on site somewhere.
Something else to consider using for that is a router (but a pre electric version) which is like a flat spokeshave handle with a vertical L shaped blade you adjust up or down for depth of cut. Joiners used them to clean the bottoms of housings to a precise depth eg in staircases (a job done almost always with routers or cnc now) But if the blade was razor sharp and set a touch below your routed level it would give you a smooth flush finish.
The other thing that occured to me was did the viking artisans ever use stamped or embossed textures in the sunk areas of a relief carving like yours? Even simple dots or ring shapes look good.