I'd RECORD in 24 bit/44.1 for the tracks going to the next studio. The BB/RT files will be fine as they are. Most modern DAWs are 32 bit floating point so whether importing 16 or 24 files - it's not the issue perhaps it once was.

When you give the next studio the stems - individual tracks - leave the 24bit at 24bit and the 16 bit BB/RT ones they're going to replace at 16 bit. Don't sweat it. You won't need to convert or dither any individual wav files you'll be sending.

Lastly, if they need to change from 44.1 to 48.1 later for video or certain broadcast applications they'll handle that with no problem on their final mix masters so there's no need to worry about that in this situation.