wrote

I need to capture a half-feral cat with a serious medical problem.
I have a large wire cage that's not designed to be used as a trap.
If I tie string/rope to the open door, pass it through the wire cage
and run it along the ground, it would be possible to wait and pull
the string once the cat enters. This won't work unless the door
closes and at the same time a knot ties it shut. The knot needn't
hold long, 10 seconds against a large strong cat should do the
job. Question is, what knot should I use?

There is no such knot.

What you need is some form of spring loaded latch like you see
on rabbit or small animal leg traps. The string just triggers that.
The latch just keeps the door shut until you close it yourself, it
isnt involved in trapping the cat, the cage still does that.