
There is no problem to get someone else to run (or even fully handle) your dog. You do not need to put the name on the entry form. All you need to do is tell the steward you have a runner and can they let the judge know! runner is best then standing at the side of the ring (outside) or by the table. (I don't like it when the runner is standing next to the handler all though out the class!)
At crufts last year I was running a dog for a friend...and was only told on the day!!!

they got to the show and basically said "roxanne can you run soanso for me" lol :D
So yea you don't need to write anything down, just let the steward know and get it sorted with the runner where it will be best for them to stand!
Good luck with both the showing and the baby!