Can weak batteries cause an ABS control module malfunction?

it is possible for low voltage to cause all your problems a would start by replacing bad batteries first then run diagnostics if you still have problems

Full charge of the battery is around 12.6 volts. I feel like one of them may be low for you, as I would not expect the differentiation between the two batteries when running. They should be load tested for accurate results. If they pass, a charging system test is in order.