Assuming you can mock store state and fully render your application it should be doable with puppeteer/selenium or similar, but this would likely mean mocking out HTTP calls to return immediately or some alternative to hijack XHR calls and return a particular payload
It sounds pretty high burden, can you focus on testing the critical path instead?
Definitely cypress [0]. I can't recommend selenium based solutions if you care about execution speed and don't care about the browser.