There is enough in the literature to extract a result of this form, but it might not appear explicitly. I will reference recent work of Thorner and Zaman instead of Lagarias-Odlyzko, since it gives a substantial improvement:
Let $L/F$ be a Galois extension with group G, and let $C\subseteq G$ be a conjugacy class. Let $H\subseteq G$ be an abelian subgroup such that $C\cap H\neq\emptyset$, let $K$ be the fixed field of $H$, and choose $g_C\in C\cap H$. Let $Q$ be the largest norm of the conductors of the Hecke characters in the dual group of $H$. Assume that $\zeta_L(s)$ has a Siegel zero $\beta_1$. Since $\zeta_L(s)$ factors as a product of Hecke characters over $K$ in $\widehat{H}$, it follows from Landau-type arguments that $\beta_1$ must be a real simple zero of exactly one such Hecke character having order $\leq 2$. Call this Hecke character $\chi_1$. There exist absolute and effectively computable constants $c$ and $c'$ such that if $x\geq (D_K Q [K:\mathbb{Q}]^{[K:\mathbb{Q}]})^{c}$, then
$\pi_C(x) = \frac{|C|}{|G|}(\mathrm{Li}(x)-\chi_1(g_C)\mathrm{Li}(x^{\beta_1}))\Big(1+O\Big(\exp\Big[-\frac{c'\log x}{\log(D_K Q [K:\mathbb{Q}]^{[K:\mathbb{Q}]})}\Big]+\exp\Big[-\frac{(c'\log x)^{1/2}}{[K:\mathbb{Q}]^{1/2}}\Big]\Big)\Big)$.
A Siegel-type argument will give an ineffective lower bound on $L(1,\chi_1)$, leading to the ineffective bound $\beta_1\leq 1-c(\epsilon)(D_K Q [K:\mathbb{Q}]^{[K:\mathbb{Q}]})^{-\epsilon}$. (Maybe this is not explicitly in the literature, but I doubt that, and if I'm wrong, it is standard to prove once you've seen the usual proof of Siegel's theorem. The biquadratic extension that you'd construct would now lie over $K$, not $F$ or $\mathbb{Q}$. The convexity bound that one would use in this argument was worked out by Rademacher.) This should give you exactly what you are looking for when you solve for how large $x$ should be for the "Siegel contribution" to be absorbed into the second error term.
As far as effective bounds go (per GH from MO's question in the comments), I do not know of any effective bound better than Stark's that holds in complete generality. One can do better if, for instance, the root discriminant of $K$ is not too small. Stark's bound accounts for what happens in infinite class towers, where the root discriminant is fixed and the degree blows up. If one could rule out such a possibility (which might be inherent in the fixed degree assumption in the original post), then Stark's bound should experience a dramatic improvement.
ADDED: The Siegel-type bound for the exceptional zero follows from work of Fogels. When the order of the exceptional character is 1, the bound is in Lemma 16 of "On the zeros of Hecke's L-functions. I, II." (Acta Arith.). When the order is 2, the bound is in "Über die Ausnahmenullstelle der Heckeschen L-Funktionen" (Acta Arith.).
Thorner, Jesse; Zaman, Asif, A unified and improved Chebotarev density theorem, Algebra Number Theory 13, No. 5, 1039-1068 (2019). ZBL1443.11239.