Unfortunately the "new" voltage regulator can certainly be the problem, if it's a typical parts house part, it would buy another to validate the issue persists. Ensure the VR has a good ground. You should also check that with the "key on", the VR 12v wire does not have excessive voltage drop. You should see minimal voltage difference from the battery to the VR, otherwise it will think the battery is lower than it really is [typically not by 4v+ however].