I looked into this problem again, and the Application.Addins collection seems to have all the addins listed in the Tools->Addins menu, with a boolean value stating whether or not an addin is installed. So what seems to work for me now is to loop through all addins and if .Installed = true then I set .Installed to False and back to True, and that seems to properly load my addins.
Function ReloadXLAddins(TheXLApp As Excel.Application) As Boolean
Dim CurrAddin As Excel.AddIn
For Each CurrAddin In TheXLApp.AddIns
If CurrAddin.Installed Then
CurrAddin.Installed = False
CurrAddin.Installed = True
End If
Next CurrAddin
End Function