I wouldn't see why not if it's just wood. The smaller the chunks the faster it decomposes.

I don't know the size of the compost bin or pile nor the size of the smoke box. As long as you have nitrogen rich greens it will work on the wood a brown or carbon material.

If you screen the compost before using it any chunky bits of charred wood would go back into the compost for more time to decompose.

You could. However, in my experience, charcoal in a compost pile takes a very long time to be digested. I have even found bits and pieces of charcoal, several years old, in various planting beds.

You may want to leave out the ashes though, as they're very alkaline.

Apparently char can be a great addition to soil, the finer the grind the better.

Yes, it's great for the compost. The goal is not to break it down but instead let it soak up all that it can from that compost. When you plant, that biochar goes into the soil and holds nutrients that it's absorbed for a long time.

Biochar is made at 500 centigrade by pyrolysis, in the complete absence of oxygen:

https://www.researchgate.net/publication/315663038_Biochar_a_gardener's_primer_WSU_Extension_Fact_Sheet_FS147E

Charcoal from a home grill, and charred wood, are not biochar. There is way too much oxygen in such an environment.