Is it really necessary to work with the fraction field here?

No, in this method of proof of Eisenstein's Criterion we need only that if $D$ is a domain then $X$ is $\rm\color{#c00}{prime}$ in $D[X]$ (by $D[X]/(X)\cong D$ is a domain, or directly by by examining lead coefs), and the further key idea that products (so powers) of $\rm\color{#c00}{primes}$ always have unique prime factorizations. So $\,a X^n = f g\,\Rightarrow f = b X^j,\ g = c X^k,\ a=bc, n = j+k.\,$ The uniqueness claim has an obvious inductive proof using Euclid's Lemma, here using $\,X\mid fg\Rightarrow X\mid f\,$ or $\,X\mid g,\,$ same as in $\Bbb Z$.

Note that the uniqueness proof works even if the domain $D$ is not a UFD (that other irreducibles may not be prime, or that some elements may have no factorizations into irreducbles does not affect this specific case - see the related idea at the heart of Nagata's Lemma).