No... and yes.
Balloon looks for Part properties, so don't waste too much effort making them look for anything else. Leaders are generally the same - so neither option does what you need.
However, this old post shows how you can do what you ask: Solved: Re: Reference solid body name in balloon text? - Autodesk Community - Inventor
Since the post is approaching the Autodesk 10-year cull policy, here (for the next 10 years hopefully) is code.
Note - you need to place balloons on the drawing of the multibody (they'll all show the same value). Next select the balloons one at a time and run the rule. The balloon text should update to reflect the body name.
Sub Main()
Dim oSS As SelectSet
oSS = ThisApplication.ActiveDocument.SelectSet
oSS.Item(1).BalloonValueSets.Item(1).OverrideValue = GetSolidName(oSS.Item(1))
End Sub
Function GetSolidName(oBalloon As Balloon) As String
Dim oLeader As Leader
oLeader = oBalloon.Leader
Dim oNode As LeaderNode
Dim oModelGeom
For Each oNode In oLeader.AllNodes
If Not (oNode.AttachedEntity Is Nothing)
oModelGeom = oNode.AttachedEntity.Geometry.ModelGeometry
Exit For
End If
Next
If oModelGeom.Type = ObjectTypeEnum.kEdgeObject
oSurfaceBody = oModelGeom.Faces.Item(1).SurfaceBody
Else
oSurfaceBody = oModelGeom.SurfaceBody
End If
GetSolidName = oSurfaceBody.Name
End Function

Peter