I'm not sure I'd use anything inside the tuner bores in the peghead, as the wood shouldn't hang on to the smell so much, but you could use the baking soda paste on a toothbrush, wipe it off. Baking soda is real easy to work with.
Possibly light use of an essential oil in places like tuner bores, pickup cavities? I use incense therapy - Nag Champa - on guitars and cases that smell like drier sheets, can't handle that smell, it's made me sell some guitars.
I actually like the smell of musty old guitars/cases, much prefer it to some "fragrance" grown in a lab.
Yes, the handle/hardware rust points to a ferrous metal, mild steel probably.
Gorgomyte leaves a nice finish on "raw wood" the fingerboard, I don't oil it afterwards, but F1 Bore oil is fairly highly recommended for fingerboards, has a strong smell you might like that should help cover other smells.
They claim it's 100% natural, just oils from trees, but it smells real "pretty", but somehow not so gross that I don't mind getting it all over my fingers working it into the board, wipes off nicely, leaves no residue, and the smell does fade away like something natural.
I wouldn't put any oil in the tuner bores, except maybe a touch of 100% pure essential oil - no carrier oil - on a Q tip, as excess oil in that area is reputed to possibly rot the wood, or make it soft or something.