I have TIs on my P and nickel rounds on my BB1025X. Never tried flats on the BB since it sounds so good with rounds, but it you have two of them, you should definitely go for it.
TIs are great, they're not as floppy as people make them out to be. A lot more going on in the midrange than traditional flats, but you can always roll the tone off to make them sound more old skool. I don't like the sticky feel and high tension of Chromes and LaBellas. Pyramid flats are also a great string, but again, not cheap.
Others that I've not tried but get a lot of love are the GHS Precisions, Fender, Sadwosky and Lakland flats.
[i]A note about stringing through the body....[/i]
On my P bass, the break angle of the strings going over the bridge saddles looks something like 30 degrees, going over a nice smooth round barrel, where it has an inch or so of length before it encounters the sharp angle going through the body.
On the BB, the strings sit on a sharp fulcrum followed by an immediate 45 degree angle. On the P Bass, the sharpest angle is at a fixed position on the base of the bridge plate, but on the BB, the sharpest angle is on the saddle, so I would be very wary of fitting any flats through the body on this bridge, especially if you need to adjust the intonation after fitting them (which you will). Thankfully you can still string through the bridge, I think this would be safest option!
Stringing through the body is supposed to increase sustain, which you don't really want with flats anyway!