First, the LM7812 (is LM1728 a typo?) is there to be a switchable regulator. It's output is set by R8 to provide a high charging current. As the battery's terminal voltage approaches this voltage, the charging current drops. This is sensed at R5, and the LM7812 is switched to a lower voltage output, set by R7, that is used to maintain a trickle charge to the battery. The voltages at the output terminals were selected from data sheets for SLA batteries and adjusted to maintain a charging currents required by this circuit.
You could replace T1, D1, and C1 with a regulated supply, but it would need to be 18vDC or greater, to overcome the U1's minimum I/O differential and the Vf drop of D6. Removing the LM7812 would defeat the whole point of this circuit. The circuit switches from high charge current to trickle charge. If you want to do something with your existing regulated 16V power supply you will need to find a different approach. Adding a diode to your original circuit's output (as in my D6), and having the relay cut the power to your regulated supply would probably do what you envision.
This was just an offering of the design of a commercial charger I'm currently using.
Ken