The ever-swaying tight-rope we walk in larp between “yay, it’s really real!” and “woohoo, I can do things I can’t in real life!” is a tricky one indeed.
A potential way this could be balanced, is to have spells represented by categories, rather than one specific origami-fold. For example, “magic shield (1 magical AP) is cast with a bird folded from red paper”, or “crippling lifeleech (drain 2HP/minute for 10 minutes) is cast with a turtle/tortoise folded from black paper”.
This gives the caster a lot of freedom to express their character through the style and choice of their folds, and also encourages mages to talk with each other, and share ideas, learning easier/faster/more beautiful origami folds.
For combat situations, you may wish to fold that bird as fast as possible, using a big piece of paper & a really simple fold to make it easy to get your shield up in time. But if you were going to give a gift of protection to a loved one, you may labour for an hour over a small piece of ornate red paper, creating a unique & intricate bird that truly expresses the intent & thought behind the gift.
To provide arbitration (i.e. so I can’t just hold up a triangle of paper and say, “It’s a dragon!”), a “master mage” GM/NPC could be present, to whom lesser mages could present new or non-standard folds for inspection. The master could then either congratulate the caster (“Ah, you have taken another step towards the 12th-folded enlightenment!”), or points out a “problem” with the fold (“Oh no, these vertices have tainted your invocation - destroy it quickly for it imperils you!”), to indicate whether it will be considered legal in the game rules or not.
The “master mage” could also teach/train casters when they gain new spells, helping them with the basic fold for the category. This, to me, feels way cooler than just reading in your character sheet “Oh, and you can now cast summon greater sandwich”.
Just throwing some more thoughts on the idea-fire 
[size=85]*edited for further clarity[/size]

