I have tuned 3 Fender Jazz basses CGDA and it worked fine on all of them. With that tuning you only have to widen one slot on the nut but I don't think it is any harder to do four slots on one bass than one slot on three different basses.
You can use the Circle K and D'Addario string tension charts to build a custom string set that will give you the same tension you have now or any other tension setup you would like to try. BassStringsOnline.com can sell you a custom string set for many brands, Circle K will sell you their strings. But since you are tuning BEAD you can just use the bottom four strings from a five string set to get reasonable tensions.
You might want to consider getting a 24 fret bass though. When you tune BEAD you lose five half steps off the top of your range to get the extra five at the bottom end. People vary, some find that to be a major issue, some find it to be a non-issue. Since you seem to be a 20 fret Fender guy at the moment you can gain four of the five back by using a 24 fret bass for BEAD tuning.