TL;DR: Origin won't let you do this by itself. Consider a mod organizer (which you can still run through Origin if you really like Origin that much).
There are some fundamental misunderstandings we need to clear up before we can answer the question. Adding a "non-platform game" to a platform does not make a seperate instance of the game, it simply creates a shortcut. In OP's case, the Skyrim install Origin is pointing to is the Steam install. There's no difference. Now we can move on to mods.
Generally speaking, there are two kinds of mods for Skyrim: mods that overwrite game files, and mods that don't. The most down and dirty way to mod a game is to do the former, simply replacing a Bethesda-made asset with a identically named file. Mods like this fundamentally alter the install, because the original file is gone. Other mods use .esp files to modifiy the game. .esp files are added to the load order and are run when the game starts, and can be toggled off to return the game to its default state. In my experience, most popular mods are of the second type, but there are exceptions. Additionally, there are mods that add to or change Skyrim's animations, which require the user to run a separate tool, usually FNIS, to work properly. At least in the case of FNIS, the tool must be run whenever a change is made, even when disabling or uninstalling animation mods. In OP's case, since mods work the way they do, and Origin is just using the Steam install, there's no built-in way to automatically turn on mods when Skyrim is launched via Origin. We need another piece of software.
For anyone doing any amount of modding to Skyrim, I cannot recommend getting a mod organizer (such as MO2 on Nexus) strongly enough. This especially applies in OP's case, as OP can use the mod manager to run Skyrim with a modded load order, while the other person can just launch from Steam using the default load order. Mod organizers streamline the downloading and installation of mods, allow easy management of the load order, and can let the user know when a mod is out of date or trying to overwrite files. You can even add shortcuts to vital tools like TES5Edit, LOOT or FNIS, so everything is in one place. Some even allow the creation of profiles, allowing the user to switch between present selections of mods in a dropdown bar. Mod organizers are standalone pieces of software and as such can be run straight from the desktop, but can also be added to platforms such as Steam or Origin if the user prefers.