What an action-packed house yours must be :)
The hair-dryer-and-external air plan needs a hell of a lot of dry ice, because you're constantly drawing in "warm" (much warmer than the dry ice) air, then losing the cooling by venting cold air. So it looks like you have 2 options:
1 dry freezing - set up a fan (like DCarch says, a personal battery fan ?) inside the box, and freeze using circulating cold air. You can vacuum-pack the meat first, or leave it naked and wrap afterwards.
2 wet freezing with (high proof) alcohol. Test your vacuum-seal plastic in the alcohol first - vacuum seal something absorbent so you can open it after and see if there was any seepage. Then if the result looks good, put enough depth of alcohol in your foam box, chill it by "dissolving" a load of dry ice, and drop in your sealed steak. Make sure the steak is on a rack so the alcohol can at least convection flow.
Unless you can get a strong fan, my guess is #2 will give the faster freeze. Either way, smaller pieces of steak will freeze quicker than one larger piece (surface-area-to-volume ratio), so break the meat down into the sizes you'll finally use it in, first.