Hello, I'm Dr. Deb and will do my best to help you today.
I'm sorry for this concern for Lucy.
It's quite possible that she ingested part of the TBone which has irritated the back of her throat. I'm not a huge fan of coincidence but there could be possible other explanations for her coughing such as tracheobronchitis (aka kennel cough), bronchitis, collapsing trachea (common in these smaller breeds). Cardiac issues can also cause coughing but seems less likely unless she also has a heart murmur.
I might feed her soft food or kibble soaked in water for 10-15 minutes over the next few days to give her throat a chance to heal. In most cases, if the throat is just irritated from something that was eaten, then the coughing stops pretty quickly.
However, if it continues and/or she becomes lethargic and/or she doesn't want to eat, then a vet visit might be prudent.
If you might be worried that she has something stuck in her mouth, a quick peek in there should reassure you; this doesn't actually happen very often. And, if you think something might be caught in her throat (which is even more uncommon), give her a biscuit or treat to eat. If she swallows it with no issues, then it's highly unlikely that she has an obstruction. Most dogs with either one of these problems will paw at their mouth or make odd motions with their tongues in an attempt to dislodge the object.
I hope this helps. Deb