Definitely keep 'em as-is! Those seemingly extra attachments are invaluable when executing the Bottle-Opener Gambit!
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Modifying chess pieces?
My theory is that Bundesform, introduced at the chess Olympiad in Munich 1936, and meant to replace Staunton pieces by a German design did not only omit the Christian cross and mitre, but also opted to bring in dogs instead of horses. Look at the greatly illustrated munich 36 Bundesform thread in this section with lots of styles to find at least 5 sorts of dogs.
I see your set in this tradition, sporting a sled dog design for a knight, somewhat between husky and laika. Now that you see the fur, you won't cut it off, or will you?
I’ll report back if I try to modify them. Once they’re set up and I get used to seeing them around maybe this quirk will grow on me.

If it represents the fur of a sled dog, it is a reference, continuing the line of knights resembling dogs of the original bundesform sets. I rather have a reference instead of a glossed-over modern version of a historically loaded set containing a noble Zagreb style knight or a neutral plastic toy horse head. But freeing the horse in the dog diy I could not object to, either ![]()
One of the first four or five bundesform dog knights: 
Bundesform is no favourite of mine, but it prevailed in tournament play for more than half a century in Germany and discarding the style would not serve understanding.
https://royalchessmall.com/products/bundesform-german-staunton-alternative-chess-pieces-set-golden-rosewood-3-8
I don’t know why they thought to add that part that looks like the horses mouth opening up (is it supposed to be the bridle?). I’ve never seen anything like that. If I knew what it was supposed to be/based off of, it wouldn’t bother me so much.
Since I got the set on a whim anyways, I was thinking about the risk involved taking a dremel to it so the ‘throat’ is the typical curve leading to the body.
Has anyone ever done anything like this? I’m aware I’ll need to refinish it with something like paste wax.