What about a digital modeller?
A Zoom B3 would be perfect, it has not only a DI XLR on the back, but it's also got a USB socket that you can use as a recording interface on your PC.
If you play guitar as well, you could look at something like the Line6 HD500x. It also has DI outputs, but it also has the USB socket, and can be used for recording. You will need to buy the HD Bass pack, but it is a great all round tool for guitar & bass. Sounds good too (yes, I have one and use it mainly for the bass).
The Boss GT-10B does the same thing, and there are others that are equally as capable.