Like my Collect Bodies mods in previous Bethesda games, crouch to get the prompt to collect the body. It will show up in the Misc tab of your pip boy, with a weight calculated by the weight of the creature plus anything it still had on its body. Dropping it from the Misc tab will return it to the world.

Requires F4SE and all DLCs to work, non-negotiable.
Hard limit of 3000 bodies. Could you REALLY need more? lol
Extra functionality requires MCM but it's not a hard requirement.

Collecting Bodies
To collect a body, kill the npc and crouch in front of the body.
You should see an interaction key to Pick Up the npc. Often it is Space, aka Spacebar.
Using that will place the npc in your inventory with the same name it had in the world.

The npc should no longer despawn naturally. This will contribute to your save file size, but it shouldn't be by much.
In order to get rid of a body, remove the Preserving Salts in its inventory. This should allow it to despawn.

Corpse Jars
To use, craft them in Workshop mode, exit Workshop mode, and activate the Jar. This will open its container.
Place ONE of your collected corpses inside, and exit the container menu.

The body will be placed next to the jar. Follow onscreen instructions to place the body inside the jar, which will have no collision(you can walk through it).
On dropping the body, you will be asked if you want to accept the position. You can retry infinitely.

When you accept the position, the jar will become solid and the body should remain inside.

TO UPDATE
Uninstall the mod and make a clean save
1. Drop all currently collected bodies on the ground or use the MCM to purge them all.
2. UNINSTALL the mod, run the game and make a NEW save.
3. Exit the game, then install the new version of the mod.
4. Run the game with the NEW version, and pick up the bodies again.

COMPATIBILITY
Any mods that add perks to interact with dead bodies are likely going to have conflicts unless they specifically require you to be standing.