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As a workaround, evaluate a rational function at zero to get its constant term as a number field element.

Below some musings around the question, including the workaround already suggested in the question.

Define the number field, the polynomial ring, and the field of rational functions:

sage: x = polygen(ZZ)
sage: F.<u> = NumberField(x^2 - 3)
sage: R.<y> = PolynomialRing(F)
sage: Q = R.fraction_field()

A constant rational function:

sage: a = Q(2*u)
sage: a

Membership tests, with one surprise:

sage: a in Q, a in R, a in F
(True, True, False)

Forcing a into F does not work:

sage: F(a)
Traceback (most recent call last)
TypeError: unable to convert 2*u to Number Field in u with defining polynomial x^2 - 3

Going through the polynomial ring:

sage: b = R(a)
sage: b
sage: b in Q, b in R, b in F
(True, True, True)

Evaluating at zero to get the constant term as a base field element:

sage: c = a(0)
sage: c
sage: c in Q, c in R, c in F
(True, True, True)