How to find a specific chest

I think it is around a week or so. That's roughly the amount of time it took to restock for me, anyway.

On PC, you can find its ID with the help "Item" 0 command, then use player.additem ######. A nearly complete list of all items with their IDs can be found on the Unofficial Elder Scrolls Pages [UESP] Wiki.

I know you "prefer" not to do that, but it's your only option.

Console players cannot retrieve or spawn items that have disappeared and have generally a hard time getting back persistent items (like quest items or unique quest rewards) which have fallen through the many cracks of the world, though the details of this depend on the specific item and installed DLCs.