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Instead of using the symbolic ring, you can view x0 as an undeterminate in a polynomial ring (or in fact, a fracion field):

sage: R.<x0> = PolynomialRing(QQ)
sage: R
Univariate Polynomial Ring in x0 over Rational Field
sage: F = R.fraction_field()
sage: F
Fraction Field of Univariate Polynomial Ring in x0 over Rational Field
sage: S.<x,y> = PolynomialRing(F, order='lex')
sage: S
Multivariate Polynomial Ring in x, y over Fraction Field of Univariate Polynomial Ring in x0 over Rational Field
sage: I = S.ideal(
....:         x**2 + y**2 - x0**2,
....:         x + y - x0
....:         )
sage: I.variety()
verbose 0 (2088: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
[{y: 0, x: x0}, {y: x0, x: 0}]

However, i am not sure if this will scale to your concrete examples (it could be nice to provide them), since the "field with constants" you are working on requires that we use a generic slow implementations for the computation of groebner bases.