Yes – the bi-interpretability works far below full PA, and it has in fact been written down. One convenient place to find it is in

Richard Kaye, “Models of Peano Arithmetic”, Oxford Logic Guides 15, Oxford Univ. Press, 1991.

Section 14 (“The field of fractions of a discretely ordered ring”) contains

• Lemma 14.4: for every discretely ordered ring (M\models Q) (i.e. Robinson arithmetic; this is strictly weaker than (I!\Delta _0), hence weaker than EFA) the usual quotient construction that turns the multiplicative monoid (M!\setminus!{0}) into its fraction field can be carried out inside (M) by a definable equivalence relation on ordered pairs.
• Lemma 14.8: Julia Robinson’s single first–order formula (\iota(x)) that defines “(x) is an integer” in the field of rationals still defines the initial ring inside every fraction field (Frac(M)).
• Theorem 14.9 (p. 244): for every discretely ordered ring (M\models Q) the structures (M) and (Frac(M)) are bi-interpretable.

Because the proofs use only the axioms of Robinson arithmetic together with rudimentary (∆₀) reasoning, all the more they go through for every model of the stronger fragment (I!\Delta _0+\mathrm{exp}) (EFA). Thus:

Theorem (Kaye, 1991). If (M\models I!\Delta _0+\mathrm{exp}) then its field of fractions ( \mathbb{Q}^M = Frac(M)) is bi-interpretable with (M).

Further sources that record essentially the same fact (again working in theories no stronger than EFA) are

1. Ali Enayat, “Classical arithmetic fragments from a semantic point of view”, in “Perspectives on the Foundations of Mathematics”, Fields Institute Comm. 41 (2014), §6.
2. Richard Kaye & Tin Lok Wong, “On interpretations of arithmetic in fields”, Ann. Pure Appl. Logic 102 (2000), 173-188 (Prop. 2.2).

So the strengthening of Theorem B with (PA) replaced by (EFA) is not only true; detailed proofs are already available, with Kaye’s monograph providing the clearest stand-alone exposition.