Original mesh showing on LOD 0

Share your package and blend files. Possibly it's a cut number issue. 

Can’t check the files right now but that is not what one needs to do. Check this thread sims4studio.com/thread/13120/cut-numbers-easy-explanation

When a part of the item needs to have a different shader assigned to it or interact with other items in a specific way, it is separated from the rest of the mesh and tuned differently.
In the package these parts are called Geometry, you can see a number of them in the warehouse tab.
When you export them from the package, s4s assigns numbers to them. The first Geometry gets 0000, the second one 0001, and so on.
Before importing the blend back, you need to make sure that these numbers correspond. You need the custom part in the calf area to replace the original part in the calf area so that it gets the tuning meant for calves, and so on. So whatever that has 0000 will replace the first geometry, 0001 the second, and so on.

My point is: clone an item that is similar to yours, export the blend, split yours the same way and assign corresponding cut numbers. Do this for each lod.

CAS packages can be weird. Sometimes, the only way to get the geometry resources and/or the texture resources to show up in the Warehouse is to export the mesh (or texture) and then immediately reimport the same mesh (or texture).

What Mauvemorn was explaining were just things you need to know in general. Many times, a CAS mesh or an object mesh will be split up into more than one mesh group because one mesh group needs a specific shader (e.g. part of the mesh needs to be transparent in the game) or a mesh group needs to be tuned in a special way to fulfill a certain purpose (e.g. pants being tuned to work with high boots).

Necklaces have two mesh groups: the necklace, itself, and that weird square that's hidden inside the sim's chest. I do not know what that box is for. mauvemorn do you know?

The child's strawberry necklace has a cut number of 0001 and the little box is 0000. If you made a mesh from scratch and you wanted to replace an EA mesh with your mesh you need to make sure the cut numbers match the EA cut numbers of whatever you cloned for your final package. In your case, you're just converting a child's necklace to an adult, right? So, export all of the LODs of the child's necklace (there are four), export all of the textures, as well: diffuse, shadow, specular, normal (bump map).

Clone an adult necklace, export all of its LODs. Open the adult necklace LOD 0 in Blender. Append the child's LOD 0 (File > Append > navigate to the LOD 0 > Object > hold shift and select s4studio_mesh_1 and s4studio_mesh_2 > Append from Library). If you get an extra rig—it will show up as rig.001—right-click it and choose delete. A rig is needed for animation so it's important but you should not have two. Delete the bone_bone_shape that will more than likely get appended, as well.

Delete the adult necklace you cloned. You can keep the box thing since it's already located where it should be and just delete the box thing that came with the child's necklace. Select the child's necklace and use G to move it upward and Z to constrain your movement to the Z-axis. You can enter edit mode and press S to resize the necklace. R to rotate. When rotating, it helps to be in right view mode (Numpad 3) or left view mode (Ctrl + Numpad 3). Get the necklace into the position you want and the size you want. Make sure the necklace is assigned the same cut number as whatever you cloned for your final package. Probably the necklace will be 0001 and the box thing 0000. Save your file. This is your LOD 0.

To make the lower LODs you can reduce the polycount of the LOD 0 using decimate, which will wreck the topology so be aware of that, or enter edge-select and use shift + Alt to select edge rings, then X > dissolve edges to get rid of edge rings.

If you choose to do the above, then delete the rig and all of the body parts within the rig. Delete the box thing, too. All that should be left in Blender is the necklace that you've resized and repositioned. Reduce the polycount as explained above, then save. This will be your LOD 1. Reduce the polycount some more. Save. This will be LOD 2. Repeat for LOD 3.

Close Blender and reopen it or start a new scene. Open the adult necklace LOD 1. Delete the necklace. I don't think there will be a little box for LOD 1, LOD 2, or LOD 3. If there is, keep it. Append the child's LOD 1 that you made. If an extra rig and/or bone_bone_shape are appended delete them. Your child's necklace should already be the right size and in the right position. Give it the same cut number that the adult necklace had. Probably, it's 0000. Do the same for the other LODs.

Alternatively, you can just open the Adult LOD 1, delete the necklace, append the child's LOD 1, then resize and reposition the child necklace, assign the correct cut number, then save. Repeat for the other LODs. I don't like this method because it's hard to get the necklace the same size and in the same position as your LOD 0.

You will need to adjust the shadow texture. This texture is used so that it looks as if the necklace is casting a shadow onto the sim's neck. It keeps things looking 3D instead of 2D. You can read more about it here and here but if you have more questions don't hesitate to ask!

Import your LODs, import the diffuse, specular, bump map, your edited shadow, make whatever adjustments to the catalog tags. Done!