Ideas for building a foldable rake
I look at those parts, and I think that your broomstick (which might work fine for a broomstick) will probably snap in fairly short order when used with a rake.
A rake handle has much more significant bending forces on it than a broom handle does, in normal use.
If the detachable rake comes with a handle, I would cut that in sections to fit in your backpack, and then I would cut sections of hardwood to be bolted across the joints, pretty much the same length as your handle sections (not shorter pieces) and drill holes to bolt them together with wingnuts. If the handle is hollow, you might be able to use sections of hardwood dowel (or wooden rake handle) inside the hollow handle, but my base assumption is that it might be solid wood itself, so an overlapping bolted series of sticks is my thought.