I'd try finishing up with a soft bristled brush.

Just as a reminder, the engraving on that era snare was done by machine not laser and be mindful of the clear coat as well when cleaning/polishing (i.e., use no abrasive cleaners, avoid the ammonia based cleaners). A damp cloth follwed by an immediate drying is probably the safest method. 

Look for a lint-free polishing cloth that has an ultra-tight weave and avoid the plush or high-pyle type cloth to prevent snags you mentioned above. 

Thank you Hoppy.

However I've tried the tightest weave cloth available and unfortunately it still snags on the engraving.

I understand this problem very well, especially with the 70s Ludwig Black Beauty models with deep engraving – the black copper material and the surface is not covered with a glossy protective coating like the newer models. The best way I use is a super small paint brush (the kind used in fine art) to lightly brush in the direction of the engraving, then blow with dry compressed air. Using a microfiber cloth or suede is very easy to leave fibers, so I avoid it completely.

Machines like the Ludwig Black Beauty really need specialized tools. There are many narrow crevices that need cleaning to keep them from being affected while operating. Instructions are available to search and follow.

The last two responses are bots so disregard.

I like to use those cheap paint brushes with the tan brisles. If you cut the bristles to about 1/4" long, they will be quite stiff and are excellent for scrubbing nooks and crannies without doing any damage. A fairly strong solution of water and Dawn is good at loosening embedded crud. If you are concerned about the edge of the aluminum bristle retainer contacting anything, wrap it with a layer or two of electrical tape, just hanging over the edge of the aluminum.

Do keep in mind, that the engraved areas can become tarninshed, and you won't be able to do much about that. Just embrace the patina as part of the history of the drum.