I have imported fusama panels consisting of paper laminated to silk with a design printed on the silk. I tried gluing them to the doors and although we had it perfectly smooth as it cured the silk upper layer developed big bubbles and air pockets.
No idea how to proceed with the next try.
Doors – Gluing fusama paper
doors
Related Solutions
You can - and people do use two sheets of washi (shojigami) on a fusuma door. However, this is often because of a repair or as in your case, the remainder of a single sheet won’t fit.
The seam is almost impossible to hide because you’ll have to overlap the edges of the paper. It won’t be visible from one side but very visible from the other.
Your home center should have rolls that’ll fit most fusuma doors so if you really don’t want a seam, get a larger roll. The new synthetic washi is much more tear and UV resistant and is worth the extra money, even accounting for the special double sided tape you have to use (normal rice glue doesn’t work on synthetics).
Slightly contrarian take: start over.
(I know, you've done a bunch of work, etc. They call it the sunk-cost fallacy for a reason. Strong butt joints in the rail are possible but hard and would involve a ton of wood filler afterwards.)
Get another door. Rough cut next to the hinge stile (but not cutting off the molding on the stile). Extract the panels and cut them to size -- no difficult joining. Clean off the dregs of the rails beside the stile. Cut your rails to an appropriate width and use whatever joinery seems best. (You'll probably need to do a bit of chisel work to flatten out the molding where the butt of the rail meets the stile.) I'd biscuit for alignment, then after glue-up is dry, sink 8-10" structural screws (2 or 3 per rail). 1/2" dowels would be good as well, as long as you were drilling the holes for them when the door was glued up and flat. Trying to drill two sides of the dowel separately is a recipe for frustration and alignment disaster.
--- Edit ---
Here's a series of pics that show what I'm talking about above.
Starting point. Sorry about the rotation.
Rough cut the rail. Don't cut any of the molding on the stile.
Mark where the rail copes over the molding on the stile. You'll cut that part of the stile later.
The molding on the stile runs all the way through under the coped rail. Somehow, chip/cut away the remaining rail.
You can do part of it with a chopsaw (granted, awkward with an 82" stile and impossible for the midrail), but don't cut past your pencil line. You'll inevitably be left with some wood that the chopsaw won't touch, so chisel/ multitool/ whatever to clean that flat.
Cut the miter that you marked with the pencil earlier on the stile molding. (Don't do what I did and bust off some of the molding while demolishing the remainder of the rail.)
Cut the rail to width and miter the molding.
Your parts should go together perfectly. Join them as described above.
Best Answer
I would change the fixing method to avoid glue.
I suggest that you consider using a method of clamping the panels into place.
If the new panel is used to cover the old surface or panel of the door then a fixing trim around the edge may be sufficient. If the old panel is to be removed so the new one is to be visible from both sides then two bits of trim will be needed.