If i was your doctor i would advise u to get your x ray soft tissue
neck done to rule out if there is any piece of bone left in your throat or not. But as you said that you are able to do your normal routine work, chances of it being stuck in your throat are less likely. The discomfort that you are feeling is because of the abrasion in your esophagus
that happened because of the bone piece stucked there. It will heal with time and does not have any serious life threat. Try to eat semi solid foods till it heals completely. Drink as much as u can. Glycerine gargles will help to relieve the pain and discomfort.