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I believe I have some more insight as to why this issue was happening. Since I want to consider toric ideals in the context of a lattice polytope $P$, my point configuration matrix should have had a row of $1$'s appended to the bottom since I'm really looking at the polytope embedded at height $1$ in the cone over $P$ (${\rm cone}(P)$). I was operating in the wrong dimension! Generating the whole ring makes sense with the setup in my original problem statement since two of the lattice points are the standard basis vectors. I made this realization because the ideal should be homogenenous as the polytope given as the convex hull of those $7$ points is IDP but several of the generators contained a constant term. So, I should have had something more along the lines of

R = PolynomialRing(QQ, 'z', 7, order='lex')
z = R.gens()
IA = ToricIdeal([[1,0,0,0,-1,-1,-2],[0,1,0,-1,-1,-2,-3],[1,1,1,1,1,1,1]],polynomial_ring=R)