Miniature Perfume Bottle with Painted Wooden Head Cap

This has no maker's mark or label, but this bottle looks a lot like the vintage perfume bottles with painted wooden heads (for the caps) by Karoff.

The miniature bottle with cap is 2 ¾ in height.

I'm estimating this piece is circa 1940s or perhaps a bit earlier- 1930s- although I cannot be certain.

Please note the paint loss as seen in photos four, five and ten. There are two areas, the one near the top of the hat being the most obvious. The paint loss as shown in the tenth photo is near the edge/bottom of the hat and is speck-sized.

Thank you for looking.