Originally posted by Hammer:

try shooting the chest with a bow until it breaks

^ This is the simplest option for what you want. The personal chest has 10,000 hp so it'll take a while and a lot of arrows, but it'll work (I used to use a personal chest as a target for archery practice before an update made that option no longer viable). As long as you don't go so fast that you accidentally fire an arrow after the chest has already been destroyed, you won't have to worry about damaging anything else in your tavern.

Also, remember to stick with plain wood arrows. You don't want to unnecessarily waste resources over this.