How to pull splintered sheathing into an electrical box?
Despite your difficulties, the cable's sheath must come all the way into the box past the point of split, and be secured by the box's cable clamps.
You cannot modify the box to make things easier.
If  you look closely at the photo, you can see where all 3 cables  have crimp marks from where the previous clamps gripped them in the correct position.  You need to do the same.
If you can't make it work any other way, I suggest you obtain a metal box of similar depth, which has a number of 3/4&quot; circular &quot;knockouts&quot; on it, three on the side you need.  Then, obtain these cable clamps.

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Start by removing the large diameter thin &quot;nut&quot; on the backside of them (barely visible) - save that, you'll need it later.
Bunch the wires together for one cable, and push them through the hole.  The side you are seeing in the photo goes first, i.e. on the wall side.  Get it so at least 1/4&quot; of intact sheath is past the edge of it (fully on good sheath and then some), and tighten down those screws.
On your steel box, bust out three knockouts on the correct side.
Thread all the cable's wires together and through the hole.  Also put the nut, from earlier, over the cable's wires.   Pull the cable into the steel box until the fitting's threads come through the hole.  Spin the nut down onto the threads.
*Wasn't that easier? You got to put the clamp on the cable separate from putting the cable into the box.   And putting the cable in the box, you got to work entirely on the accessible side.
Now, those nuts need to be torqued down, so get a flat screwdriver and put the blade on the &quot;bump&quot; on the left side and bap the back of the screw driver with some force, e.g. with your fist or something .  That way you tighten the nut clockwise.  That's how you do these.
The other boxes aren't working for you because they assume you  have access to both the inside and outside of the box after it's mounted.  You don't, actually.