I have this Tumi case that is exactly the same size as the BA cage and it doesn't even get close to fitting the trays. Never had a problem at security, though a few months ago the goons at flight connections did keep insisting that it needed checking right up until the point at which I placed it in the cage. I left them silenced as I carried on my merry way (until I reached the Slow Track queue).

If it doesn't fit in the cage, however, then you'll need to either check it in or find a different bag to take the paraphenalia in.