Technique to find a mesh intersecting with a primitive solid without using colliders?

There are hundreds if not thousands of meshes with hundreds of vertices each. There is one sphere that can be intersecting any of those meshes and is constantly moving.

Are AABB colliders approximating the mesh's volume the best way to go about this? Or is there a more specific technique that can be used here?

(A limitation I have is that my colliders need to be rotated, so they're each stored in a game object, so the number of game objects becomes absurdly large for my meshes(tens of thousands) which is slowing down my game.)