Bayesians and modern "frequentists" both use the same underlying laws of probability theory, and in my view, both can reconcile their views on the "frequentist definition of probability". Both groups of practitioners agree with the strong law of large numbers, which says that if you have an exchangeable sequence $X_1,X_2,X_3,...$ then (with probability one):$^\dagger$
$$\mathbb{P}(X_i \in \mathcal{A}) = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n \mathbb{I}(X_i \in \mathcal{A})
\quad \quad \quad \text{for any measurable set } \mathcal{A}.$$
Frequentists take this as a "definition" of probability, whereas Bayesians use an alternative interpretation of probability (usually an epistemic interpretation), but also agree that it corresponds to limiting frequency in this case. Bayesians don't disagree with the strong law of large numbers, and they tend to view it as something that clarifies the notion of "repeated trials" in the frequentist viewpoint ---i.e., the idea of infinitely repeated trials of a stable experiment is provided by the concept of exchangeability.
If you go back to the older probabilists like Venn, Reichenbach and Von Mises, you will find that they made philosophical arguments about the frequentist meaning without the aid of Kolmogorov's strong law of large numbers. To do this, they tended to invoke the idea of repeated trials that could occur hypothetically infinitely often, under unchanging conditions. This is captured well by the modern condition of exchangeability of an infinite sequence, which is the basis for the strong law of large numbers.
$^\dagger$ One might wonder if it is legitimate to invoke laws of probability theory in answer to determining the meaning of probability --- i.e., whether this is circular logic. For this purpose we may treat $\mathbb{P}$ as purely a mathematical operator, and then imbut it as a "probability" only once we have satisfied ourselves that it meets the requirements of the meaning of this concept.