I might try one or two other things:

Get an aluminum tube that has nearly the same outer diameter (OD) as the hiking stick's inner diameter (ID), and use this to support the broken pieces from the inside. The aluminum doesn't have to be really stout. An old lawn chair with the right diameter tubing would work. Use rivets to hold it in place, then fiberglass from the outside as MauganN20 suggests. You also may want to first sand the stick where you apply fiberglass to remove the sheen and give the resin something rough to cling to.

Rather than using your hands to squeeze out resin (definitely wear gloves no matter what), clamp a window squeegee to a table and turn the stick against it in the direction the glass cloth was rolled on. Squeegee each layer independently, then let the first layer get tacky - not dry - then apply the second layer in the opposite direction and squeegee again. Also, wet the cloth completely with resin before applying it to the stick and skip the tape.

You'll have a hotstick that's tougher in the repaired section than anywhere else.