Do you have the width in the last photo to walk into the next space? Is it that the kitchen itself is changing after this? The wall with the hvac vent I'm guessing is up for demo next . .

For my part, covering the beam itself with drywall so that you can trim the drywall seems like a bad idea. Drywall the ceiling in full, and dry wall the walls right up to and under and besides the beam ends, use edging on the top under the beams .. but leave the wood beam exposed, stain it walnut after you sand off / (now you will have to fill holes from the drywalling with stainable putty). You are hiding some very expensive character . . looks like the rest wants to be white cabinets, muted blue green walls . ..