Does the battery voltage measure between 13 and 15 volts when the engine is running? With it being closer to 15 when the battery is partially discharged, like when it has sat in the cold outside overnight before starting the engine? And does the battery voltage measure around 12-12.5 volts with the engine off, measured after the car has sat overnight?
If so, and the voltage regulator is part of the alternator, I think what I’d do in this situation since pretty much everything is new, I’d take the alternator back to the place you bought it from and ask them to put it on their test jig, to see if it is working correctly. Most auto parts stores have a test jig for this purpose. If not, an auto-electric shop would have one.
Just a fyi, for the charging system to work correctly the cabling between the battery and alternator positive has to be very low resistance, on the order of 0.005 ohm. The only way to get that low of resistance is if every connection is perfectly clean and tight. Any corrosions or oxidation, no way will it work. Likewise for the connections between the battery negative to the chassis ground, and the alternator negative to the chassis ground. And since the alternator is probably grounded to the engine, not the chassis, the engine has to be well grounded to the chassis too. This was all done at the factory when the car was built, which is why it doesn’t cause problems in most cars for the life of the car. But if any of those connections have ever been removed, positive or grounding cables changed, etc, definitely something to suspect.
Edit: I should add one time my Corolla had similar problems, and I discovered a bad splice in the positive wire between the alternator and the battery. It wasn’t easy to find, buried in a thick wiring harness down in the middle of the engine. Not a fun job. I had to use a sharp knife to cut away the outer wrapping of the harness, then I just followed that wire as it wound its way from the battery toward the alternator. Lo and behold, there at the splice it was down to just 3 or 4 strands out of 20. What happened, the battery was leaking a little which I didn’t notice, and the battery acid was slowly decanting down that cable, since the cable was touching the battery. It followed the cable down, staying inside the harness so I couldn’t see it, until it hit that splice, where it arrived at bare metal and promptly ate the soldering gadget that joined the two thick wires. I replaced that gadget, properly re-joining the cables with a low-ohm connection, and the leaking battery, and it fixed the problem.