And, another alternative - pack the joint with baking soda, carefully wipe off/dust off all the excess, Then drip thin CA into the joint, soaking the soda. Any soda on the surface near the area that gets CA on it will set up rock hard, so only have it where you want it.
I use this technique to fill old firewall holes. It is so hard and strong, you can drill new ones into it, and even use a tap to cut threads in it.
I've even used a variation to restore rotting portions of my deck on my house. 5 years in rain/snow/sun (+100 to -20 below!) and still far better than the wood. Sounds silly, but actually much cheaper than the "wood rot restore" products at the hardware store.