OK, so here's what I mean. Get the frame seat and measure the distance X of the sling. This distance must be kept fairly precise otherwise the seat won't be level.
Now remove the upholstery and cut the frame in half.
Now you need to find some pipe to serve as a sleeve. It could be PVC or aluminum or lightweight steel, it just needs to slide over the frame fairly snuggly. Then cut the sleeves and mount to the bottom of the toilet seat with 8 spring clips. Make sure the sleeves are separated by that same distance X.
Now you can slide the frame into the sleeves.
Once it is all assembled, drill holes from the underside through both the sleeve and frame pipes, near the end of the frame pipe (you won't be able to see the end, so you will have to measure where it is). You need to mark all the parts so you can reassemble it the same way each time. Then remove the frame and put spring pin inserts into ends.
You should probably drill each hole and add the spring pin before moving on to the next hole. This will keep everything aligned while you are drilling. When you are done, you can cut the seat in half if you want.
I don't think you will even need a hinge. If the spring clips are strong enough, everything will stay fairly secure. This should disassemble into a fairly compact package. If you want it even shorter, cut off the bottom of the frames and add leg extensions using the same sleeves and spring pins. If you don't keep the legs the same length that they were, then you might have to play around a little with the front/back lengths in order to make the seat level. This is also a good method if you couldn't maintain that distance X. Now you can take the leg extensions off, and nothing will be too long or wide.