Bellow I would explain how I would deal with that situation:
Accept the fact that you will have to refinish the whole stock in order to achieve a good, quality repair. If you are satisfied with sloppy work, one that's just for repairing that stock to function without care for the looks, you might skip the bellow text.
You have a big piece with several cracks here, so a CA glue might not be the best solution as you will not have the time to properly apply it to all of the surfaces and press the piece to blend in with the stock. You simply risk to leave a raised piece that is not completely glued - small patches of glue here and there. Instead use a good quality waterproof wood glue, but NOT the fast curing one - I'm quite satisfied with Bison brand wood glues, but they might be tough to find in the US. Lift gently the cracked piece with a thin blade and insert generous amounts of glue with a syringe. After you're done press hard on the piece several times with your fingers so to squeeze out all the excess glue. Clean all of the excess glue with a wet rag, thoroughly - don't worry that the edges will not be covered with glue, you want that and you will deal with that later. This is actually why I use wood glue - I can easily remove all of it from the surface so I can fill the cracks properly later.
Put the stock in a machinist vise, level it, cover the piece with some thin nylon foil (old grocery bag, food wrap foil...) and put weights on top to press the piece leveled with the stock. Surgical tubing might not be the best solution, as the cracked piece is on a flat surface and the tubing can't apply sufficient force to press it down. Leave it to cure for 24 hours. Sand the stock and proceed to fill the cracks with your finishing media of choice. You may use shellac - apply small amounts of it into the cracks and sand it gently with 600 grit sandpaper with a backing. Repeat until the cracks are all filled and blended with the surface. Instead of shellac you may use TruOil - in that case leave it to cure until it starts to feel tacky and proceed with the sanding. I don't like to use steel wool for that application - it leaves fine steel particles and it will contaminate the seam.
After you are satisfied with the result, apply wood stain (optional) and finish the stock with whatever you like - BLO, TruOil, polyurethane...
Proceed to have fun!