The description for the item Harkon’s bite goes like this:

A dire wolf tooth dangles from this simple cord necklace. While you wear it, the necklace grants you a +1 bonus to ability checks and saving throws.

this is the same as the luck stone without the curse if you read the full item description.  But luck stone increases initiative were as Harkon’s bite does not.  It’s supposed to since initiative is an ability check but on dnd beyond it doesn’t.

You're right, Harkon's Bite should add +1 to initiative.

Thx for the fix