First, I would spray. BUT, I know there are circumstances where that simply isn't possible, or feasible.
Second, I am guessing that the bifolds are closet doors - yes? Finishing inside a dark closet (yeah... go ahead... take a light) to get the back sides of doors just isn't practical. It's too easy to goof your surface up on bifolds, and getting in and out of a closet while the finish is still wet is certainly one of them. Not to mention the practicality of it. Also, with the doors in the closed position it is very, very easy to get a run or ridge on the closed panels where they meet. Those would come off.
As far as a regular door to a room, I still take those off to finish them. It is too easy for a wet door to get messed up, catch dirt, dust, etc. I don't have any problems painting/finishing the upper parts of the doors, but don't like being on my knees to finish the bottoms. if it is one or two doors, I will leave them on. If I have to turn out the work, I take everything out to a garage, sunroom, painting area and have at it.
This is a time where an oldie but goodie will shine for you. Remember Deft brushing lacquer? It is easy to apply with a brush, pad, or foam (if you get the right foam!) and since it has a longer open time you can do a nice job if you are quick about it. The advantage here is that you can get TWO coats easily on your materials in a day. I have used this stuff on everything you can imagine including inside bathrooms and never had a problem.
With this many doors, you could set up and start sealing them, and by the time you finished it would be lunch time or after. Starting with the ones you began with in the morning, you can easily come back for a second coat after lunch. To get all 22 doors finished, you will probably be doing this for a couple of days (at least) if you are brushing.
Since you are using pine, it can still be quite resinous around gnarly areas, and can cause clouding in any finish. If this Deft is your choice, they make their own sanding sealer, but Zinsser sanding sealer works just fine. As with any finish, you need 2 coats on top of the primer to provide good protection.
Are you sure you can't spray? If the weather was warm I could put all three coats of finish on in a long day, and hang them the next. Hand applying would probably take me 3 full days to get all three coats on everything. Maybe more...
I don't know why you would wipe anything. Wiped finishes take too long to build up to get to abrasion resistant finishes. To get to an industry standard of 1 - 3 mil thick cured finish (depending on the material) you would be wiping plain old pine for days and days. You just can't get enough finish on the material per coat, and I don't have a clue how many coats it takes to seal up something porous like pine. And remember, you will have to warrant your work, so that means water resistance as well as abrasion resistance. That means a good, thick, durable finish.
Don't let your client talk you into poly/varnish/whatever he wants. With a 6 - 8 hour recoat time for those finishes, stay in the lacquer family. Poly, etc. is simply impractical in a production application like this. You will be pretty busy as it is getting all 22 doors (44 sides!) finished with the correct appearance without waiting a whole day to recoat.