Unless I'm missing something, it seems like the proof that Met and Met_oo do not have regular subobject classifiers should be simple: as usual, the underlying set of $\Omega$ would have to be in bijection with the regular subobjects of 1. In both categories, certainly the empty subset and the full subset are regular subobjects. So, whatever positive distance you give between the two points of $\Omega$, the image of $\top : 1 \to \Omega$ is clopen, so any pullback would have to give a clopen subspace. But in both cases, it's easy to find contradictions, such as the subspace $[0, \infty)$ of $\mathbb{R}$ given by the equalizer of the identity and absolute value functions $\mathbb{R} \to \mathbb{R}$.
For PMet, on the other hand, I seems that in fact ${ 0, 1 }$ with distance 0 between the two points should give a regular subobject classifier. That boils down to observing that any regular monomorphism is an isometric and injective function, so it can be expressed as the pullback of the characteristic function to $\Omega$ and $\top : 1 \to \Omega$. (In fact, that should also show that isometric and injective functions are exactly the regular monomorphisms.)
Unless I'm missing something, it seems like the proof that Met and Met_oo do not have regular subobject classifiers should be simple: as usual, the underlying set of$\Omega$ would have to be in bijection with the regular subobjects of 1. In both categories, certainly the empty subset and the full subset are regular subobjects. So, whatever positive distance you give between the two points of $\Omega$ , the image of $\top : 1 \to \Omega$ is clopen, so any pullback would have to give a clopen subspace. But in both cases, it's easy to find contradictions, such as the subspace $[0, \infty)$ of $\mathbb{R}$ given by the equalizer of the identity and absolute value functions $\mathbb{R} \to \mathbb{R}$ .
For PMet, on the other hand, I seems that in fact${ 0, 1 }$ with distance 0 between the two points should give a regular subobject classifier. That boils down to observing that any regular monomorphism is an isometric and injective function, so it can be expressed as the pullback of the characteristic function to $\Omega$ and $\top : 1 \to \Omega$ . (In fact, that should also show that isometric and injective functions are exactly the regular monomorphisms.)