I just throw my bird in and spatchcock later. Haven't brined one that way.
Opening it up like that does help some, but I can't say how much or how little. My guess would still be a minimum of 36 hours. The size of the bird will matter.
If you cut it fully into parts, you could reduce it, but I think I still recommend in 101 (I'll have to look) that breasts alone need 24 hours.
I have a bucket I always use for brining. I've got my timing down for a two day brine, so I just brine them whole and spatchcock later.
Once I have my brine flavors, my timings, etc, I hate to experiment for turkey day.