Thank you Al for your answer and sorry for the (very) late reply...
It turned out my BB734A was defective. The difference in volume between the first two strings and the last two when both PU were on was simply too much. I sent it back to the shop that recognized the problem and gave me a full refund, which I immediately invested in buying exactly the same bass from a different shop. This one works perfectly. When both PU are on there is only a slight decrease in volume in respect to having only the P pickup on, but the balance between the strings is perfect and the tone is simply amazing. I guess I could say, in your words, that I get the P tone with some added grit.