This is a photo of a surged edge being readied for a miter seam on a landing. The back of the surging has been slathered in latex so that when the final cut is made
through the surging material, it will stay together. The latex is allowed to dry for a bit before making the cut.
This cut was made at 45 degree angle. Yours will be at a 90, so this may not be necessary for you. If it is, a very thin smear from a hot glue gun at this point might work.......... test on an inconspicuous area to see if the heat melts the surging material. You might not even need to bother with this step.
The second image is the cut edge of the surging.
After you make the cut, a decent size dab of latex in the center of the surging can be gracefully dabbeled around the
inside edges of the cut with an awl to make sure all of the threads stay put. You may have to coax any loose or partial cut edges of the thread back into place........the latex will lock it in.
You need to be careful to only let adhesive get on the interior edges of the surging threads, because you don't want the adhesive to show.
Let it dry....... speed it up with a hair dryer, but don't rush this part because the two tacky edges with latex on em' will act like contact cement as you test fit the new piece.
Like Daris mentioned......... use a new blade for the cut, then immediately
and carefully seal the cut edge with latex.
Once you can handle those sealed edges safely, you can butt together the old and new surged edges and staple the new piece in place.