First you use two phrases "can be constructed out of" and "complete generating set". I think I understand what you mean here but it is possible there is a misunderstanding.
I was hoping you might get a better answer as the following is really a strategy and not a complete proof.
Take the category generated by your tensors (I hope we are in agreement as to what that means). Then this is a rigid symmetric category. In fact it is what Deligne calls a tannakian category together with a fibre functor. Then Tannakian reconstruction reconstructs an affine group scheme which has this category as its representation category (I think in the physics literature they would refer to Doplicher-Roberts).
There is a discussion of reconstruction at
Then if this is not $F_4$, what is it?
P.S. Do you need the volume form? I have not seen this in this context.
The paper http://arxiv.org/abs/0907.0256 proves a quantum analogue of the analogous result for $G_2$. The reference  in this paper is to
which proves a more general result. This result applies to the representation you are considering. This implies the quantum analogue of the result you want. The volume form is not needed.
This is probably not the simplest way to prove the result you want.