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You can use Singular's notion of a ring with parameters, but Sage's PolynomialRing interface to Singular doesn't understand this yet, so currently you have to do it by hand like so:

R = singular.ring('(0, p0, p1, p2, p3, p4, p5, p6, p7, p8, p9)', '(z0, z1, z2, z3, z4, z5, z6, z7)', 'dp')
I = singular.ideal('p0*z0*z1 - z2*z3', 'p1*z4 - z3*z5', 'p2*z1*z5 - z6', '-p3 - p4 - p5 + z4 + z5 + z6', '-p5 - p6 - p7 + z0 + z3 + z4', '-p3 - p8 - p9 + z1 + z2 + z6', '-p7 - p8 + z0 + z2', '-z6 + z7')
singular.option('redSB')
gb = singular.std(I)
S = singular.ring('(0, p0, p1, p2, p3, p4, p5, p6, p7, p8, p9)', '(z0, z1, z2, z3, z4, z5, z6, z7)', 'lp')
J = singular.ideal('fglm({}, {})'.format(R.name(), gb.name()))
J

It takes a few minutes but finally returns:

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z6-z7,
...

Better yet would be using the libSingular interface.