Personally I would use “would ring”, but it’s not something that would be wrong either way and the meaning wouldn’t change very much either way. The difference is whether you are allowing for the idea that the bell might not ring. This is phrased like this is an on going request, has asked vs had asked, so the bell ringing is a future event, something we have limited knowledge of. We expect the bell to ring but the future is not certain, so we can show this uncertainty by saying “ would ring” or ignore it. Either way, it don’t effect heart of sentence, the repeated requests, so it probably doesn’t matter.