It sounds like the camera is going into standby mode, which I don't think you can physically alter in the menu settings. One way to test whether it is indeed going into sleep mode, thereby causing an AF lag and not some other issue, is to set the metering delay mode from the default few seconds to say 5 minutes or longer. Sit the camera down for 2-3 minutes like before, then press the AF-1/BBF.
If there is no delay, like there was before, then you know it is indeed standby mode kicking in and causing the issue. The metering delay will get around this problem but, be aware that your battery will drain faster, as the camera will constantly meter for the duration you set the timer for. Not so much a problem if you have multiple batteries or access to a power outlet but, if shooting out in the field all day, you could run out of juice if you don't have enough batteries.
If there is still an AF lag, something else is afoot. Is it the same for all lenses? Are you using AF-S or AF-D lenses? You could try setting the AF to the default shutter release and see if the lag persists, or temporarily assign another button as the BBF.
Give the above a try first then let us know.