I will preface this by saying I am no expert. I have done a bit of guitar work in my time, and ran a music shop for a year. Guitar wood repair has a lot of parallels to wood repairs on rifles.
One caution with CA glue......it can really "haze" an existing finish, and turn it milky looking. I believe the fumes can do it as well as the glue itself. Of course, that may just be the finishes I worked around, and not longrifle finishes. Test before proceeding.
A trick I learned to fix a crack was (if you can get to the end grain or end of crack) is drill with a pin-vise and jeweler's drill a tiny bore hole along the center of the crack. I would then basically fill the tiny bore with "Zap-A-Gap" gap-fill super glue. That stuff will bond anything to anything......except wax paper. By filling the bore it wicked into both sides of the crack. Makes for a sturdy repair. Zap-A-Gap is used in hobby models and such, and bonds materials while filling any voids or gaps.
Again, test it around the finish you are working near. CA is some funny stuff.
I have also corresponded with folks that use Hot Stuff inside the inletting of longrifles in high-stress areas to prevent cracks. I am guessing it glues adjoining wood fibers together much like plywood.
Keep in mind that Super Glue CA does not age well, and sometimes makes a slow-curing mess if it gets too old. On anything I care about, I always crack open a new bottle.
My one last thought would be that I would try to use superglue AFTER staining, but BEFORE finishing. That way stain could still get in, but a final finish would not get hazed or milky.
God Bless, Marc