I'm only putting this out there because I have a now 21 year old picky eater.
Does this sandwich have to look a certain way and only have certain things on it? Because while those CS places may have something like a ham sandwich it may not be two pieces of white bread with deli ham between them.
I have read the sandwiches at Starring Rolls are pre made in another kitchen so there are two issues, one you can't order one with only ham and bread and two they can run out of them maybe around 2:00 PM.
For TS, yea they may be able to make you a ham sandwich on white bread but only if they have white bread and only if they have ham, which they may not. There was a similar thread that asked about grilled cheese and everyone thinking, how hard is grilled cheese? It bread, cheese and butter. Well very few places have white bread on hand so they use whatever multigrain bread or bun they have and the gruyère cheese they use for their one and only sandwich.
I just don't want you to assume that you can get a ham sandwich anywhere and have a hungry cranky kid. (Been there done that).
I like the idea of packing at least one and keeping it a cooler.