Transfer response from remote server to content script (Chrome extension)

What I want to achieve: Upon clicking icon of my Chrome extension, a file with downloaded from remote location and the user is presented with saveAs dialog.

Caveats: I'm using latest manifest from Google, namely v3.

I have edited this post many times as I I was able to achieve more and more. I leave only my latest code.

tl;dr Now almost everything works.Crucial thing is missing: response from server (body) is not saved. Instead string [object Object] is saved.

// when icon is clicked
chrome.action.onClicked.addListener(tab => {
  if (tab.url.startsWith('http')) {
    post({url: tab.url})
    .then(async res => ({
      filename: getFilename(res.headers.get('Content-Disposition')),
      blob: await res.blob()
    }))
    .then(response => {
      console.log(response);
      chrome.scripting.executeScript({
        target: {tabId: tab.id},
        func: saveFile,
        args: [response.filename, response.blob],
      })
    })
    .catch((error) => chrome.scripting.executeScript({
      target: {tabId: tab.id},
      func: showAlert,
      args: [error],
    }));
  }
});

function getFilename(header) {
  return /filename="(.+)"/.exec(header)[1];
}

async function post(data = {}) {
    return await fetch('http://localhost:5000/citation',{
        method: 'POST',
        body: new URLSearchParams(data)
    });
}

function showAlert(error) {
    let message = error.error !== null ? error.error : error.message;
    alert("Error: " + message);
}

async function saveFile(filename, blob) {
  let link = document.createElement('a');

  let url = window.URL.createObjectURL(new Blob([blob],
      {type: 'application/octet-stream'}));
  link.href = url;
  link.download = filename;
  link.click();

  // For Firefox it is necessary to delay revoking the ObjectURL.
  setTimeout(() => {
    window.URL.revokeObjectURL(url);
    }, 250);
}

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