Hm, no you can not become a block producer if the biggest stake holders don't support you. That is as certain as death on a delegated proof of stake blockchain.
Moving the goalpost. You talked about BT's vote. I am talking about BT's vote. Others' vote doesn't have a bearing on this. You can become a top 20 witness without BT's vote. It is just going to be hard.
Of course, you are not going to become a top 20 witness when multiple big stakeholders don't support you. Why does someone who doesn't have support become a top 20 witness?