fix: keep install assets inside a non-root base

Manifest start_url, scope and icon src resolve against the manifest URL,
so root-relative values sent an app served under a path prefix to the
domain root. Make them relative, and add a build test under
base: '/example/' that asserts every emitted link and manifest URL stays
inside the base. index.html is unchanged: Vite already rewrites its
links under base.

CHANGELOG.md

@@ -5,6 +5,8 @@
   instead of a blank tile. The icons are generated from `public/logo.svg` by
   `node scripts/generate-icons.mjs` and committed, so a plain `vite build` needs
   no image toolchain. The existing SVG favicon still serves the browser tab.
+  Manifest paths are relative to the manifest, so a build served under a path
+  prefix installs and launches inside that prefix.
 
 - GBFS v3 station names use English translations when available, then the first
   usable translation or short name. Empty or malformed names no longer appear as