This is related to this question:
Freyd's Geometric Finiteness : An Example Computation
I've essentially reduced the problem to the following question:
Equip $\mathbb{N}$ with the discrete topology and let $P$ be the sheaf of germs of functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(k)=0$ for all but finitely many $k\in\mathbb{N}$ and such that $f(k)\leq k$. Is $P$ K-finite?
If anyone has an insight, I would appreciate it. Also, I'm not sure if the logic tag is necessarily appropriate here, so feel free to remove it if it isn't.
Upon reflection, I want to retract this question. It seems I've confused myself by confusing "internally K-finite" with "externally K-finite". Internally it's clearly true that it is K-finite, and externally it's clearly not: If we look at the subalgebra of $Sub_{Sh(\mathbb{N})}(P)$ that contains $\emptyset$ and is closed under adjoining singletons we'll only get the "bounded" subobjects. It seems that in the paper referenced in the linked question, Freyd must have had "internally K-finite" in mind.