The way I see it, Lottie doesn’t want her imitator to know that she’s left town, so she doesn’t want “Charlotte Grote” to be seen boarding a train. Once she gets wherever it is that she’s going, if she hears about reputation-ruining antics, it’ll be easy enough to prove that she’s wherever it is she’s going, and couldn’t have gone it (I’m sure there’s more to her plan than that, but that’s likely part of it).