Bill Tindall, E.Tn.
This repair is on an antique that may today, or someday, have considerable value. I would want a very good reason to treat this repair any different from what the conservation staff of a museum would do. The current wisdom is to make repairs that can be completely undone. What museums have found is that they can not anticipate the reasons that something might need to be undone 100 years hence. And, they face undoing stuff that was thoughtlessly done 100 years prior. Hence, they use hide glue for this kind of repair, tinted shellac to add color as necessary, never stain, and finish with shellac.
If this bird was from Ikea I would say get out the CA glue and have at it.