I am still concerned about sinking the current to the negative supply. As I said, I want to use a PWM voltage inverter, which is a very weak supply and can generate only couple of mA. If I sink the 12.5 mA to this supply, isn't it basicaly the same process as if the supply itself sourced equivalent opposite current? I fear, that by sinking too much current into this supply, the negative voltage will drop and the supply will no longer be able to go close to 0V at the output. Is this a valid concern?
It is a valid concern if you need the output to get all the way to 0.0 volts and the negative bias supply cannot support the sink current required. The LM350 requires a minimum load current of roughly 10 milliamps (worst case) whether that comes from the resistor between the output and adjustment pin or from a separate load on the output.
If your negative bias supply is not going to support the needed current, then your only choice if you want to continue to use the LM350 will be a low compliance current sink from the output to ground and a larger resistor between the output and adjustment pin to supply the current through the diodes into the operational amplifier outputs. 1.25 volts divided by 1.2k will yield about 1 milliamp through the diode which is currently forward biased. I usually run the entire LM317/LM350 10 milliamp minimum output current through the diodes into PNP emitter followers. This is also a good place to put a pair of LEDs to show if the output is voltage or current limited.
The minimum output voltage will then be limited by the load and the current sink to ground. With no load, is a minimum output voltage of about 0.3 volts enough? How close to 0.0 volts do you need?
You could use a different regulator (1) which does not have such a high minimum current or replace the regulator with a discrete transistor stage. Personally I like using integrated regulators because of their built in protection circuits.
While not the power supply I would emulate exactly, the well documented Tektronix PS501 and PS503 use discrete pass transistor output stages and *still* pull a current to below ground so that they can achieve a 0.0 volt output with good stability. The PS503 actually deliberately regulates to a few millivolts below ground to account for the worst case offset voltage of the error amplifier.
(1) I did a quick search and did not find any direct replacements for the LM317/LM350 with a lower minimum output current.