I've done that several times with a smaller enlarger (5x7), so I could get twelve 35mm frames, or six 6x4.5.
As the proof sheets are just proof sheets I do not need them be perfectly aligned.
Use gloves. Might sound silly but I do most of darkroom printing without them and even if I do, I usually use just one. But when I put a few strips at once into a big carrier I definitely use both.
I put the strips perpendicular to the edge where upper and lower parts of carrier are joint together. Wearing gloves I can hold the other ends of the strips and begin to close the carrier.
If the film is quite flat it sometimes is quite straightforward. Film do not usually move much if I close the carrier slowly. If it moves when there is no room for fingers i can move the ends using a toothpick or straightened paperclip.
If the film is not to flat the almost closed carrier holds the other end somewhat with my fingers still holding them. I move away fingers and close more. Usually they do need a toothpick...
Rather tedious so I gave it up.
I do contacts with sleeved negative so you might try putting sleeved negs into the carrier.