Belmopan is closer to Mexico.
Here's why: Belmopan is the capital of Belize, which sits directly north of Guatemala on the Yucatán Peninsula. Guatemala City, capital of Guatemala, lies further south. Since Belize shares a border with Mexico (Quintana Roo), while Guatemala does not, Belmopan is geographically nearer to Mexican territory.
Rough distances to the Mexico border:
So if you're measuring proximity to Mexico, Belize's capital wins decisively.