After lots of help from others, I realized the question boils down to some sort of stable reduction, which is a properness statement for the moduli of stable or relative stable maps $\overline{M}(X/V)$.

The isomorphism $C_K \simeq D_K$ provides a point $C_K \to C_K \times D_K$ over $K$ of $\overline{M}(C \times D/\text{Spec} R)$. Properness (=stable reduction) extends this to a map $C' \to C \times D$ over $\text{Spec}R$. The maps $C' \to C, C' \to D$ must be partial stabilizations because that's a closed condition in the base, and they're blowups because they're proper birational.

The blowups are at closed subschemes supported at a finite number of points in the special fiber, and I wish I could say the centers are smooth or something (or iterated blowups at smooth centers), but I don't need it.

Can anyone see why that would be, say by the fact that the exceptional divisors are chains of $\mathbb{P}^1$'s or something?

I apologize if this question and answer were obvious to the broader community.

After lots of help from others, I realized the question boils down to some sort of stable reduction, which is a properness statement for the moduli of stable or relative stable maps $\overline{M}(X/V)$.

The isomorphism $C_K \simeq D_K$ provides a point $C_K \to C_K \times D_K$ over $K$ of $\overline{M}(C \times D/\operatorname{Spec} R)$. Properness (=stable reduction) extends this to a map $C' \to C \times D$ over $\operatorname{Spec}R$. The maps $C' \to C, C' \to D$ must be partial stabilizations because that's a closed condition in the base, and they're blowups because they're proper birational.

The blowups are at closed subschemes supported at a finite number of points in the special fiber, and I wish I could say the centers are smooth or something (or iterated blowups at smooth centers), but I don't need it.

Can anyone see why that would be, say by the fact that the exceptional divisors are chains of $\mathbb{P}^1$'s or something?

I apologize if this question and answer were obvious to the broader community.