For a demo recording at a high priced studio, I don't know how practical it is to do it the way I do it. But for the sake of answering your question I'll tell you how I like to do it.
I use four channels. Yes, four. Two direct lines two mic'ed lines. The first in line is a direct right out of the bass, usually this is the best box they have in the studio - mostly just trying the sound of each one and deciding on the purest and mostly obnoxious sound of the strings on wood sound does the trick. Then out of that box the signal goes through all my effects and amp head. Then the second direct box is hooked inline out of the back of the head, the back of my Ampeg SVT5 Pro has a nice balanced output that gives a really clean sound but also gain from the preamp and with all the eq of the head and the effects together. Then out in the live room the SVT 8x10 cabinet is mic'ed usually with a RE20, that's the huge mic with all the slats in it that is used to enhance the low end. This mic is placed a few feet from the cab to give the long low sound waves some room to travel. Then right smack up again the speaker is usually a Shure 57 or 58. Not the best mic in the world, but a studio standard that doesn't color the sound much. Then each signal is pumped through the board and blended however the sound fits best. Sometimes having one or both of the direct faders up more works, sometimes the cabinet mics higher than the DI lines is the better tone. Each studio is different and each session is different so there is no real one way to do it, just mainly what tone fits the other instruments and the overall mix the best.
But naturally I haven't done every recording this way. Sometimes the studio isn't fitted to accomdate this way. Sometimes I travel far and don't have my amp with me so I rely a lot on a good DI sound and hope the mixing engineer can re-amp the direct signal. Thats where they send your already recorded part out to a cabinet in the live room and mic it like you were there playing it on an amp, and then it gets a little time shift and mixed right back in the song.
Having to choose one or the other is always a compromise. A direct alone lacks balls and a mic'ed amp alone lacks attack and clarity. There are some amp simulators that work sometimes, but to me it still sounds "fake" and I cringe if the engineer isn't a good illusionist. I know guys that go in the studio with only a Line 6 Bass POD and are happy with their sound. I've only tried that once, but I'm pretty sure that time I was re-amped during mixdown.
Once again, if it's a demo recording it's not that important - not that I ever want to take importance away from a good tone. But the budget and facility are always a deciding factor. But if you can, run a direct box right out of the bass and combine the signal with the mic'ed cabinet. This can be done on one channel if the engineer is a 'closeminded who cares about bass' guy. Which there are more of than not.
So yes, I do use a DI (direct box), but only for part of the sound. Hope that helps you out. Good luck, and remember not to give in to someone else's routine of overlooking a great bass tone. It will help your case during mixdown. A good take, and a good tone will be audible. If it isn't - fire the engineer and remix it!
SDG