Normally to repair a simple split like this in wood that will go together so neatly I would suggest a PVA adhesive (e.g. Evo-Stik blue, or Everbuild 502) because of the easy water cleanup of excess glue.
But there are some problems with using PVA here. With waterbased glues only fresh surfaces bond properly due to this effect and because you don't have easy access to the crack surfaces you can't do much to reliably refresh them.
So, I think the adhesive of choice becomes epoxy, despite it being a little trickier to work with in this situation. Epoxy offers two main advantages, it seems to more reliably bond old wood surfaces and also it doesn't require the very high clamp pressures that PVA does to achieve its strongest bond. Additionally if you need to glue the brass into the wood it allows you to do the whole job with the one adhesive.
I think you're going to want a use an epoxy with a long working time for a comfortable, stress-free glue-up, so regrettably inexpensive 5-min is out. Even 10 minutes may not be enough so I'd recommend 30-min epoxy at minimum. Other than cost there's no downside to using an epoxy that sets slower than that if it's all you can find as I would recommend leaving the glue to set for a day at least regardless1.
Some quick tips:
I think the ideal 'clamp' here isn't a clamp at all, but tape. Electrical tape is great for this kind of job because it has some give and resists snapping so you can stretch it tightly as you wind it on, each wind applying more and more pressure. Work from the narrow end of the crack towards the open end.
Don't disturb the stick as the glue is setting. Even a slight shift can significantly impact bond strength. So after you've finished glueing put the stick where it won't need to be moved out of the way.
Additional stuff I'd recommend: gloves so you don't get epoxy on your skin; a screwdriver for leverage; a hair dryer2.
Apply the glue. Be liberal but don't go crazy. You want to get it into the entire crack, which is what the screwdriver is for — gently lever the stick apart to open up the crack a smidge so you can get glue as far into the narrow end as possible. With local warming it should seep in well.
Now close the crack and squeeze the stick firmly. Excess glue will squeeze out. Wipe off. Squeeze again and wipe again and then it's time for the tape.
1 The epoxy's full strength may only be achieved after 24 hours or longer at normal room temperature (slower if cold), so don't go testing the bonds the following morning!
2 Use the hair dryer to warm the wood, not the epoxy. You'll be amazed at how the epoxy becomes more fluid once it touches the warmed surface, allowing it to spread and penetrate a lot more easily. If you warm the puddle of mixed epoxy it makes it too fluid to transfer easily, and significantly shortens the working time.