144 visa-free eligibility

Unfortunately it is not. The main rule within 144 hour visa free transit says that only the immediate points, regardless of the nature of your stay there, before and after your Chinese destination matter. In other words, your immediate points make your itinerary to look exactly:

Hong Kong - Beijing - Hong Kong

This is considered to be a return journey and not a transit. You are advised to reschedule your itinerary by, for example, introducing Macau after Beijing so that your itinerary looks like this:

Hong Kong - Beijing - Macau

After you have reached Macau, you can easily reach Hong Kong and from there proceed to Taiwan.