Welcome to my world of spending ages thinking, planning, doing, then realising down the line that things don’t line up. This keeps us mentally agile. Real problem solvers. Would be so boring if things went as they should.
What I would do (which looks like the option mentioned) is get a hood minus fan, fit one of those turbo 140mm in line fans to your ducting and then lay square/rectangular section ducting on top of your wall cupboards to outside. Looks like the fan may fit in the chimney opening between the beam. Ducting up the chimney is asking for gunk to roll back down.
What I would do (which looks like the option mentioned) is get a hood minus fan, fit one of those turbo 140mm in line fans to your ducting and then lay square/rectangular section ducting on top of your wall cupboards to outside. Looks like the fan may fit in the chimney opening between the beam. Ducting up the chimney is asking for gunk to roll back down.