A note for anyone having the same issue I was:
After adding the feet collider to prevent sticking to walls, my character started falling through the ladder when they were part-way up. I noticed this seemed to be happening because my feet collider was smaller than the collider for my ladders (the climbing tilemap collider). When my character’s feet collider was completely intersecting the ladder, the collision would stop registering and he would fall straight down.
The solution was to change the “Geometry Type” of the Composite Collider on my climbing tilemap to “Polygons”. By default it’s set to “Outlines” which makes the whole tilemap collider act like an edge collider. Setting it to “Polygons” makes the collider solid, and the collisions were properly registering even when the colliders were completely intersecting.
Hope this helps anyone having the same issue!