A question about uniformly bounded semigroups

Let $A$ be an unbounded linear operator of domain $D(A)$ defined on a Banach space $X$. Suppose that $A$ generates a $C_0$-semigroup $T(t)$ which is uniformly bounded. I would like to know if there are additional assumptions under which the following conjecture is true:

For every compact subset $K$ of $X$, there exists a constant $C_K$ such that $$\mid{((T(t)-T(t'))x\mid \leq C_K\mid t-t'\mid},\quad \forall t,t'\in \mathrm{R}_+,\quad \forall x\in K.$$

If I understand your question correctly, this would mean that the function $$t\mapsto T(t)x$$ is Lipschitz continuous, which is equivalent for $$x$$ to be in the Favard space $\text{Fav}(A)$\$. See for example Defintion 8.2. in these lecture notes.
If your space $$X$$ is reflexive, then the Favard space is exactly $$D(A)$$, the domain of $$A$$, see Corollary II.5.21 in
Hence if $$D(A)\neq X$$ (which is the case for unbounded generators), this fails.
Unfortunately, in non-reflexive spaces, where $$D(A)\subset \text{Fav}(A)\subset X$$ still holds but the Favard space generally fails to be the whole space. Hence, your property cannot be fulfilled even for $$K=\{x\}$$ singletons if $$x\notin \mathrm{Fav}(A)$$.