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Wait, never mind. I just remembered that this is a homogeneous curve in 3 variables rather than a projective curve in 2 variables, so the integral points of this curve are the same as the rational points of the projective curve. So it doesn't matter that EllipticCurve_from_cubic rewrites it in Weierstrass form.

A.<x,y,z> = PolynomialRing(QQ, 3)
phi = EllipticCurve_from_cubic(x^3+y^3+z^3-6*x*y*z)
C = phi.codomain()
P = C.gens()[0]
P2 = phi.inverse()(5*P)