144 visa valid for travel between hong kong -> Beijing ->hong kong

Of course 144 hours transit without visa does not apply to you. You need a visa. That’s a visit to China and not a transit.

Indeed, you are not allowed to do that because you are arriving to mainland China from area A (Hong Kong), staying in mainland China, that is area B, but you are not departing from the mainland China to area C, but to area A (Hong Kong) which is not allowed.

I would suggest you that you travel from Hong Kong (area A) to Beijing (area B), and then from Beijing to Macau (area C) from where you can easily reach Hong Kong via a newly built bridge or by using a catamaran.

Or even better you can fly like this:

Singapore - Beijing

Beijing - Hong Kong

Hong Kong - Singapore

That woould be perfectly within the legal frames for 144 hours free travel.