So, quest items don't add to your carry weight. This is great. But I'd like the option to make a given specific quest item at least seem to add weight, preferably by being able to add / remove a quantity of invisible weights to my inventory. Any suggestions on a way to do this? Alternatively, is there some way to prevent the player from dropping an item besides setting it as a quest item? The ''Can't Drop'' flag doesn't accomplish that.

Queue

ModAV CarryWeight is another solution.
In *theory* scripting items with OnDrop blocks can be used to stop them being dropped, but every time I've tried it's caused crashes on occasions when the player's gear is removed from them via RemoveAllItems.

You can add invisible weights by adding a number of unplayable Armor items to the player's inventory. Just uncheck "Playable" and give it a small weight, like 0.05 or 0.01 so it can be fairly precise. Then all you'd have to do is determine how many should be added/removed when the item is added/removed--along with a small check to ensure that they're not removed prematurely by outside means.