Just for fun, I set up the following scheme:
 A 6j symbol is everything that fulfils BiedenharnElliott. (Plus symmetry, orthogonality etc. if that doesn't follow from it anyway.)
 There are only a finite number $i$ of irreps (including $1$, the 1dim "neutral").
Then I tried out which of the many possible ClebschGordan expansions are mutally consistent with all the 6j equations via BiedenharnElliott. For two irreps I got only this: $1\bigotimes{X}=X (X=1,2), 2\bigotimes{2}=1+2.$
 Ha!
 Even I did see this before, it's the smallest Fibonacci fusion category. Now a quick research gave me a load of adjectives that come with fusion categories. Can you tell me which of them apply to my scheme?
 $i=3$ produces 3 "minis", with $i=4$ I get 6 (and I stopped here because of the dreaded combinatiorial explosion  are these already classified somewhere?).
 My main question, though, is whether you get a free parameter when you choose $i$ large enough. All my "minis" give only a set of fixed complex numbers for the values of the 6j symbols, the quantum dimension and the writhe normalizer (the minis all seem to be knottheory compatible). The latter is a root of unity so I can speculate all the minis correspond to special values of the Jones polynomial. Or suchlike.



Fusion categories are discrete (like finite groups) and you never have a "free parameter". This is an observation by Ocneanu. 


The number of nonisomorphic simple objects is called the rank of the fusion category. You've made an error somewhere, as in rank 2 you should also get an example where the nontrivial simple object squares to $1$. (In your notation $2 \otimes 2 \cong 1$.) For rank 2, Ostrik gave a complete classification (http://arxiv.org/abs/math/0203255). Ostrik's argument is somewhat indirect (via the Drinfel'd center). As far as I know, no one has given a direct classification in rank 2 via 6j symbols. The hard part is figuring out why there's no fusion categories with the fusion rule $X^2 = 1 + n X$ for $n>1$. For rank 3 if you only look at fusion categories that give knot invariants, then Ostrik has a complete classification (http://arxiv.org/abs/math/0503564). A great source of all information about "minis" that give knot invariants is Section 5.3 of RowellStrongWang (http://arxiv.org/abs/0712.1377). "Modular" means that you get knot invariants plus you have a certain "nondegeneracy" condition that's a bit hard to explain. 

