How to properly use sendResponse() in MV3 Chrome extension

I'm playing around with Promises available in Chrome extensions Manifest V3, but I'm not getting the expected results. What I'm trying to achieve: store some data from a browser action popup, then get a response back after the data is saved. Here's what I have so far:

background.js (service worker)

async function handleMessage(message, sender, sendResponse) {
  switch (message.request) {
    case 'setTimestamp':
      await chrome.storage.local.set(message.data);
      sendResponse({
        status: chrome.runtime.lastError ? 'error' : 'ok'
      });
      break;
    default:
      sendResponse(null);
  }
}

chrome.runtime.onMessage.addListener(handleMessage);

popup.js

// Event handler for button in popup.html
async function setTimestamp() {
  const response = await chrome.runtime.sendMessage({
    request: 'setTimestamp',
    data: {
      timestamp: performance.now()
    }
  });
  console.log('setTimestamp response', response);
}

The timestamp is being saved successfully, but the response in popup.html is undefined instead of {status: 'ok'} that I was expecting. Can someone please point out what I'm doing wrong?

(Note: I'm aware you can use storage.local.set() directly from the popup and validate that the data was saved successfully by detecting chrome.runtime.lastError in the callback, but I wanted to use the new aync/await pattern above.)