Let Them Eat Cake--as Long as It's Vegan

Oh, dear!  I have some meat-eating friends I adore, and they've invited me and my meque (mejor que un esposo or better than a husband) to lunch this week--first to an Italian restaurant and then to their home for dessert.

I did tell them that I'm a vegan, but I don't think they've grasped my upgrade (downgrade to them) from vegetarian to vegan.

I wrote this as part of a longer message:

We're looking forward to seeing you this Thursday, when I hope you will let me bring a cake.  I no longer eat animal products, and that includes eggs.  I can make a cake without eggs so there won't be a problem for anyone else!

This is the response I got (also as a longer message):

We have made a reservation in the Italian restaurant for 1:45.  We were assured that they have vegetarian food in their menus.  After that we would be happy to see  you both  in our home ...We are kindly asking you not to bring anything, including the cake, because {my wife} is going to bake something. 

But I'm sure the cake will have eggs in it.  What should I do?  Bring the cake in my bag?  I don't really require cake, but I've known hosts who require that guests eat it, and they don't say, "Let them eat vegan cake."