I pretty much solved my own problem, but in case other folks want to go down this path in the future: From what I gather, the animator.FrameData.SpriteData contains objects that represent the information about the way a sprite needs to be positioned and drawn. It doesn't seem to contain a reference to the name of the bone that it is attached to by default. I edited the SpriterDotNet API to add a string variable named BoneName to the SpriterObject class in the SpriterModel.cs file, and then added "info.BoneName = timeline.Name;" to the top of the AddSpatialData method in the FrameDataCalculator class. This allows me to retrieve the BoneName from a SpriterObject.
sds_alex got a reaction from Mike at BrashMonkey in SpriterDotNet - An implementation for all C# frameworks