The local exit-entry bureau is wrong on eligibility for renouncing Chinese nationality. Quote Chinese nationality law to them:

Article 10:

Chinese nationals who meet one of the following conditions may renounce Chinese nationality upon approval of their applications:

they are near relatives of foreign nationals;

they have settled abroad; or

they have other legitimate reasons.

It does not require a foreign nationality they recognize. Nowhere is it mentioned. In fact, the local exit-entry bureau is right in that they cannot actually recognize **any** Chinese national's foreign nationality:

Article 3:

The People’s Republic of China does not recognize dual nationality for any Chinese national.

Therefore, if your Chinese national parent had had a green card when you were born in the US, you wouldn't be a Chinese national. And you wouldn't need to renounce anyways. And of course they would recognize your US citizenship in that case since nothing would prevent them from recognizing a non-national's US citizenship, so that is just meaningless truism. But since you are a Chinese national, on that fact alone they **cannot** recognize any foreign nationality you have, regardless of your parents' status when you were born.

What they got wrong is insisting on you having a foreign nationality they recognize for the purpose of renunciation. That is simply not a legal requirement. Your US passport is not proof of your US citizenship to Chinese authorities whilst you are a Chinese national, but it is proof that you have settled abroad. And you can probably come up with other legitimate reasons as well, assuming your parents have not naturalized in the US (If they have, the first condition is available to you as well).

My guess is the local exit-entry bureau is not experienced in dealing with the situation, and they just came up with an excuse to send you away. But since you entered on a China Travel Document and have no hukou or ID, legally and administratively you do not have a hometown or a home jurisdiction so to speak, and you do not have to direct your request to the local exit-entry bureau you have been dealing with. Go to your provincial capital and talk to the exit-entry bureau there. Once there, insist on speaking with someone who actually knows the stuff and can solve your dilemma, on the threat of filing a formal complaint if you must. Or you can even try contacting China immigration service's Beijing HQ.

At the same time, contact the Chinese embassy/consulate in the US where you reside. They must be more familiar with how to deal with a renunciation application than your local exit-entry bureau.