I mentioned this trick in another thread:
Pressure-pack Carby Cleaner from the local automotive parts shop is handy stuff. If you get the cans with an "extension" tube, you can get the stuff into some very tight places, AND avoid hosing down the entire stock.
Keep the crack open for a day or so to ensure there is no solvent left in the grain and apply the glue of your choice.
I have been using epoxy for decades. Recently, on another thread, Peter Laidlergave me a "heads-up" on the SERIOUS adhesives used in building / repairing wooden structural components in light aircraft. Basically, if it keeps the wings from falling off.....................
As for clamping, a simple split as described should just need "reasonable" pressure applied by a padded carpenters clamp. For rounded shapes, I use strips of rubber that have been cut from inner-tubes. You can use surgical tubing, but old tubes ara a lot cheaper. Wind the rubber around the repair, stretching the rubber as you go. It's is amazing how much compression you can get.