Unity function to access the 2D box immediately from the 3D pipeline?

As far as I am aware, there is no built in for this.

However, finding the extremes yourself is really pretty easy. Getting the mesh's bounding box (the cuboid shown in the screenshot) is just how this is done, you're just doing it in a transformed space.

1. Loop through all the verticies of the mesh, doing the following:
2. Transform the point from local to world space (this handles dealing with scale and rotation)
3. Transform the point from world space to screen space
4. Determine if the new point's X and Y are above/below the stored min/max values, if so, update the stored min/max with the new value
5. After looping over all vertices, you'll have 4 values: min-X, min-Y, max-X, and max-Y. Now you can construct your bounding rectangle

You may also wish to first perform a Gift Wrapping of the model first, and only deal with the resulting convex hull (as no points not part of the convex hull will ever be outside the bounds of the convex hull). If you intend to draw this screen space rectangle while the model moves, scales, or rotates on screen, and have to recompute the bounding box, then you'll want to do this and cache the result.

Note that this does not work if the model animates (e.g. if your humanoid stands up and does jumping jacks). Solving for the animated case is much more difficult, as you would have to treat every frame of every animation as part of the original mesh for the purposes of the convex hull solving (to insure that none of your animations ever move a part of the mesh outside the convex hull), increasing the complexity by a power.