Odd coincidence - I have recently thought through this very topic - false beams. Musings from my thoughts:
The piece attached to the joists does not have to be a continuous, full-length board. It can be short blocks - like 6" - at the joists. You just need to keep them in line when you anchor them - a full-length straight board tacked into place so that the nail holes left behind will be hidden by the sides.
Perhaps, given your close spacing, you could even use some 1/2" ply or mdf pieces - sized for the spacing between the joist blocks...........get the first row in place, then use the ply jig for consistent offset and a straight line for the next row.
The sides of the false beams don't have to be 1". They can be 1/2" - 5/8" - whatever yield you get from the bulk stock you start with. Run through the bandsaw, and then through the planer to give you consistent thickness and one good face. Then through the TS with a glue-line blade for one good edge, and consistent dimension.
The bottom of the beam also does not need to be very thick.
The full-length side and bottom boards...............I have given consideration to shorter boards, with beveled overlap joints.............but I'm not really sold on that idea.........just an option I was thinking about. Easier to handle, cheaper to buy shorter lengths...........but not the same look.
I admit that I was not considering something with spacing nearly as "tight" as yours............but I don't see how that changes my thinking...............
Looking forward to seeing where you land, and what the project looks like. Would help me with my thought process - learn from someone else's mistakes......not that I am rooting for you to have any problems, of course.
Best of luck
Kent
When I started woodworking, I didn't know squat. I have progressed in 30 years - now I do know squat.