I agree, I would use an ice pack and/or thermos. You could send hard boiled eggs, soup, etc. Does she eat apple sauce? You could freeze it into a freezer pop and maybe 'trick' her into thinking it's a treat? Ditto w/ a smoothie that incorporates greens like spinach.
Also, I don't understand the pay thing. You have to pay for their meal (?) ... presumably for food your kid isn't going to eat anyway? I would try to talk to them and ask what their protocol is. I mean they can't MAKE your kid eat something, so I would just send whatever items fit the criteria, including some things your kid will actually eat. I guess I don't really understand what the recourse of the place is. I would probably address that with them.