Simple normal crossings divisor and locally monomial functions

I figured this out after a while of sitting on it.

The solution I started to present need a little addition to it, but is largely correct. In the notation of the post,

Since $\widetilde{\varphi}$ vanishes on $\widetilde{X}_0$, it should be in $I(\widetilde{X}_0)$, which is locally principal, so I can write it as $\sum f_ig$ where $g$ is locally a generator of $I(\widetilde{X}_0)$. I can then collect terms to write the function as $h(x_1^{a_1}\cdots x_m^{a_m})$ where $a_i$ are maximal non-negative integers. Then $h$ cannot vanish on any subvarieties intersecting $\widetilde{X}_0$ by the stipulation that $V(\widetilde{\varphi})=\widetilde{X}_0$. So $h$ is locally a unit, and the proof is finished.