The expat package exists, but is disappearing. If you can get anything close to that, you are lucky. Strickly speaking, 20-30K is not in the expat bracket. It’s more like a high end local pay. But it is definetely good compensations to yield you a very comfortable living status in BJ. And reasonable for your experiences and educations.
The tax rate in China is progressive rate. The highest bracket is 45%. You should be somewhere between 15-20%. Since you have a Singapore PR, you are qualified for a foreign resident exemption of 4,000 Yuan/month (for local Chinese, I believe the exemption is 800/month.).
You housing request might be the reason, which made your HR negotiator uncomfortable. Because the housing, a western style housing in BJ is expensive. It could run as high as 2 to 3 times as your salary for a technical expat. My suggestion is you should simply ask for a straight housing allowances, say 5,000/Yuan a month. With that money, you should be able to find something perfectly fine.
I am not familiar with Singapore tax regulations. But in the US, up to this moment, the first 80K US is tax exempt anyway. An expat would only need to pay China tax. Quite a few companies offer expat tax equalizations, so that they do not pay more (or less) than they have to in the US. But I think your case would be different.