Larry Clinton At Frankfort, (Central) Indiana
I don't use CA glue except for one thing when woodworking and that is to obscure a very minor tear out in the surface. I wet the area with CA then sand, and repeat until the tear out is not visible.
Hot glue is the answer. I use my "hot pot" to heat hot glue but if your not set up to do that the liquid hot glue will work well.
It has several advantages.
First: it will work without clamping as a rubbed joint.
Second: the "beak" can be removed with heat after it is set if the alignment is not perfect.
Third: after the glue has set, if you work quickly you can remove any squeeze out with a cotton swab and water.
If the joint is not invisible, I would touch it up with a very small artist brush using either oil based artist pigmented paint blended to color match, or a stain that matches the color of the piece.