install.js

self.addEventListener('install', function(event) {
    // Perform install steps
    event.waitUntil(
        caches.open(BB_SW_CACHE_NAME)
        .then(function(cache) {
            BB_urlsToCache.map(function(url) {
                var request = new Request(url, {mode: 'no-cors'});

                fetch(request).then(function(response) {
                    return caches.open(BB_SW_CACHE_NAME).then(function(c) {
                        return c.put(request, response);
                    })
                })
            })
        })
    );
});

The promise for all the caching isn't being passed back to waitUntil, it's lost after the first .then.

self.addEventListener('install', event => {
  event.waitUntil(
    caches.open(BB_SW_CACHE_NAME).then(cache => {
      return Promise.all(
        BB_urlsToCache.map(url => {
          const request = new Request(url, {mode: 'no-cors'});
          return fetch(request).then(response => cache.put(request, response));
        })
      );
    })
  );
});