You have a valid concern that he might not be able to digest bone as it's not an item that is easily processed by the body. We often see that cooked bone causes irritation of the GI tract (including the throat, which may be why he's hacking), puncture, lacerations and blockage.
Since you've seen bloody production, my advice to you would be to have him examined and treated by your vet. Failing this, I can give you some steps to take at home to help your companion’s stomach feel better. However, if you do not see a marked improvement from your pet or you see worsening of symptoms, they absolutely must be examined by a veterinarian.
The first step is to administer a dose of regular pepcid (famotidine) every 12-24 hours. This should help with GI symptoms. You will want to give 0.5mg/pound of body weight (a 10# ***** would receive 5mg, a 5# ***** would receive 2.5mg, etc). For this, you can visit any human pharmacy and buy the OTC brand name Pepcid, or you can use the cheaper, off-brand “famotidine” that’s available. Either will be useful. For a pet avoiding taking medication readily, you will likely need to using a pilling technique like this one: https://www.youtube.com/watch?v=-P6NfbxeLX0
2 hours following a dose of famotidine, the time needed for the medication to begin working, you can offer a soft, bland diet. To make this, you’ll combine white or brown rice, boneless, skinless chicken breast and sufficient water for cooking in a stock pot. Boil on medium until it turns to mush and the breast is easily flaked. To avoid nausea, start with small amounts to begin with and offer the amount every 2-4 hours. A few teaspoons to start is typically sufficient and you can work your way up every 2-4 hours in incremental increases until you’re sure no vomiting will be seen. If your companion requires a more palatable food, try adding in pureed baby food in chicken, turkey and similar flavors. Avoid those that contain onion or garlic in the ingredient panel. Work up to feeding exclusively until at least 3 days. After this, work on slowly switching back to the regular food that your companion typically eats. The softer food should give the throat a chance to heal if this might be due to irritation alone.
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