^^^This. I have a pair of the Seymour Duncan Custom Shop '51 stack pickups in my Fender CS P. Same pickup that is in the Fender CS Dusty Hill P. Good tonal range and can get very thumpy. I've also heard good things about the Fralin replacement, but I've also heard good about the Lollar which leads me to the strings below. Check the dimensions and info prior to ordering. A search here for replacement pickups for the Sting P bass will give you good info. If I recall, some of them don't drop in without a mod. Oh, and a bonus with these is there is no hum because of their design.
Strings - I have only read half of this thread at this point, but what strings do you have on it? If you want thump then you need to go with flats, some thumpier than others.
Body Rout for new pickup type - I wouldn't do a standard precision pickup in this. No reason. The one pickup that I think is so cool in the 51-56 body style is the TV Jones as others have mentioned here. Looks great and is highly regarded. Plenty of videos/demos online for those so you can hear it.
Oh, and I vote keep it....that's a great bass and it has a lot of sentimental value to you. Once you replace the pickup and look into the strings, you'll have a big smile on your face when playing it.