Doing a pickup cavity with a router is done in no time (assuming you buy or make a template). Surely your neighbor won't go nuts from 2-3 minutes of router noise? Because that really all it takes, although if you've never used a router before it might be a good idea to do a couple of practice runs on a piece of scrap wood.
If you don't use a template... Well, that need a lot more skill, and a good quality router. Freehand routing isn't something you should do on a bass unless you know 100% that you have the skills and experience to pull it off. For a pickup cavity I STRONGLY recommend buying or making a template. Routing a pickup cavity using a template is so easy that someone who has never, ever used a router before should be able to get a good result (although I still strongly recommend practicing on a piece of scrap, to get the "feel" of it).
Of course you can also do it with a combination of drilling out most of the cavity and using a chisel for the fine work, but getting a good result with a chisel takes a lot practice. The fact that you need to ask these questions tells me that you do not have the necessary skills to get a good result using a chisel. You will also need a good quality, sharp chisel and ideally learn how to sharpen the chisel yourself, which is also a skill that needs to be learned.
My advice?
Router+template. You will be most likely be finished before the neighbours even have the time to complain.
And of course, whatever you end up doing, make sure you get all measurements right. THAT's the most critical part.
Also, I don't think you've mentioned whether there's a pickguard involved. A pickguard can hide a less-than-perfect cavity, although then the pickguard would have to look nice and be a perfect fit of course.
Do you have any specific pickup(s) in mind? If not, it might be worth considering that some pickups (like mudbuckers or Thundertrons with pickup rings) can hide a poorly made cavity. The only person who will know about the mess below is YOU.
(Make sure the pickup you use is suitable for your string spacing though, the two examples above are NOT suitable for Fender string spacing, but I've seen reports from other Talkbass users saying ThunderBLADES work ok).