how to google chrome extension modify the URL from the sync storage?

Use-case

When you log in with multiple Google accounts, google starts adding a query parameter authuser=X in Google URLs ...<googleservice>.google.com/home/dashboard?authuser=1

I want to be able to set the default value for the authuser parameter from the extension.

Problem

As a beginner with Chrome extensions and the frontend, I am unsure which API to use to update the value to smooth the experience and the URL is updated even if the user enters it manually or I click the link from another tab or program.

What I have so far

Right now I have popup with the text field it can read from the storage.sync.

Questions

1. Where the code should live? content_script or `service worker or both?
2. Which API to use?
   1. window.location
   2. chrome.tabs
   3. declarativeNetRequest
   4. chrome.webRequest

I appreciate your suggestions and help.

Since the redirection rule is simple due to being a static text, you only need the popup and declarativeNetRequest API, no need for a content script or the background script (service worker).

manifest.json:

  "permissions": ["declarativeNetRequestWithHostAccess"],
  "host_permissions": ["*://*.google.com/"],
  "action": {"default_popup": "popup.html"}

popup.html:

<input id=inp><button id=save>Save</button><div id=err></div>
<script src=popup.js></script>

popup.js:

document.getElementById('save').onclick = () => {
  chrome.declarativeNetRequest.updateDynamicRules({
    removeRuleIds: [1],
    addRules: [{
      id: 1,
      condition: {
        resourceTypes: ['main_frame', 'sub_frame'],
        urlFilter: '||google.com/home/dashboard*',
      },
      action: {
        type: 'redirect',
        redirect: {
          transform: {
            queryTransform: {
              addOrReplaceParams: [{
                key: 'authuser',
                value: document.getElementById('inp').value,
              }],
            },
          },
        },
      },
    }],
  }).catch(e => {
    document.getElementById('err').textContent = e.message;
  });
};