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This is not an answer, just a rewriting of the code in the question:
sage: R0.<q>=PolynomialRing(QQ); R0
Univariate Polynomial Ring in q over Rational Field
sage: k = FractionField(R0); k
Fraction Field of Univariate Polynomial Ring in q over Rational Field
sage: R.<x,y> = PolynomialRing(k); R
Multivariate Polynomial Ring in x, y over Fraction Field of Univariate Polynomial Ring in q over Rational Field
sage: I = R.ideal((q*x*y)^2); I
Ideal (q^2*x^2*y^2) of Multivariate Polynomial Ring in x, y over Fraction Field of Univariate Polynomial Ring in q over Rational Field
sage: I.radical()
...
TypeError: Cannot call Singular function 'radical' with ring parameter of type '<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_domain_with_category'>'
