Perhaps I am missing something, but the density of $C^{\infty}$ makes use of the weighted Sobolev norm. So, each function $\phi \in W_{2,0}$ may be approximated by a sequence of $C^{\infty}$ functions in the weighted norm. I don't see how this yields uniform convergence?

For the boundedness, the standard Soboloev embedding theorems can be used to show boundedness on $(\epsilon,a)$ for and small $\epsilon$ as the weights are non-zero and bounded here. Hence, in essence all we need to show is that $\phi$ is bounded as it approaches $0$. To this end, for each $\phi$ one can find $C > $ such that if $\phi(x) > C$ then $\phi'(x)> 0$ and $D < 0$ such that if $\phi(x) < D$ then $\phi'(x) < 0$. This will clearly lead to a contradiction if $\phi$ is unbounded.

Okay, so I have shown that the interpolant is not bounded, so the embedding seems to be the only option.

If the embedding is untrue, it requires a bounded oscillating function, from which it follows that the derivative will be unbounded and oscillate between $+- \infty$ as x approaches 0, similarly for the second derivative.

It is basically just the right continuity at 0 that is needed, but I couldn't see any way that this followed from boundedness or the density of $C^{\infty}$.