Here is one thing to try. Start with the smallest simplicial model for S¹ (the 2-simplex modulo its boundary). Take the free group in each degree (but force the basepoint to be the identity). The resulting simplicial group FS¹ is a model for ΩS²; furthermore, being a simplicial group, it's a Kan complex. Thus, we know there must be some map f: S²->FS¹ which represents the generator of π₂ΩS²; the group of FS¹ in degree 2 is not too big, so it should not be hard to write this down explicitly (I haven't tried, though.)

Of course, you really want a map S³->X, where X models the 2-sphere. Since FS¹ is a simplicial group, let X=BF¹, it's classifying space. X is a model for the 2-sphere, and I expect that if you examine it closely, you will see the "suspension" of f corresponds to some explicit 3-simplex in X, which is your model.

I'm not sure this counts as a "combinatorial model", of course.

(I have a vague memory that Dan Kan did something like this in one of his papers in the 50s. Is that right?)

Here is one thing to try. Start with the smallest simplicial model for S¹ (the 1-simplex modulo its boundary). Take the free group in each degree (but force the basepoint to be the identity). The resulting simplicial group FS¹ is a model for ΩS²; furthermore, being a simplicial group, it's a Kan complex. Thus, we know there must be some map f: S²->FS¹ which represents the generator of π₂ΩS²; the group of FS¹ in degree 2 is not too big, so it should not be hard to write this down explicitly (I haven't tried, though.)

Of course, you really want a map S³->X, where X models the 2-sphere. Since FS¹ is a simplicial group, let X=BF¹, it's classifying space. X is a model for the 2-sphere, and I expect that if you examine it closely, you will see the "suspension" of f corresponds to some explicit 3-simplex in X, which is your model.

I'm not sure this counts as a "combinatorial model", of course.

(I have a vague memory that Dan Kan did something like this in one of his papers in the 50s. Is that right?)