I dont recommend mailing a buttercream cake. I have seen the conveyor system at fed ex, and beleive me, a buttercream cake will not survive!
The only thing I can think of is using a high quality fondant and wrapping it well. It might have a chance in surviving.
I think your best bet is contact someone who lives in NJ and see if they will bake a cake and deliver it for you. Maybe someone here!
Perhaps if you did it in some kind of a cooler? I ordered some M&Ms, and they shipped it to me in a styrofoam cooler, completely covered in these frozen gel packs to keep it from melting. If you made your cake, put it into a well packed cake box, put the box into a styrofoam cooler with the gel coolers, and then put THAT into a box and shipped it . . . . maybe it would hold up. and be well protected. And if you shipped it next day. hth
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