Short answer - I would turn it down.
Long answer - If it really is that really good job you say it is, is it just another role or is a dream job for you? It's a personal choice but important for you to weigh up the job vs the benefits of J/F. I agree with others that if they're firm on everyone travelling Y, what else are they firm on re: hotels...expenses...overtime...flexibility etc...
They were quick to throw more money at it, they obviously want you - which could even allow you to push it again (either the J-clause in your contract one more time, or ask for an extra Łk or so...)
Good luck - do let us know what decision you make!