...
You're talking about it like it was an actual thing. Whatever kind of violation backward time travel is, it's not a run of the mill 'this particle cannot exist in two places at the same time' violation. For starters it's not the exact same particle, it's an older version of the same particle. If you had an atom of plutonium, waited for it to decay, then sent it back in time, and set it next to its younger self, they wouldn't be the exact same atom existing in two places at once.
If you have it working the way you say, it doesn't work at all - you broke the game before you started playing. You don't need to say 'your time travel machine won't work anymore when you arrive,' because if the rules work like that you won't arrive, the machine won't arrive, nothing will arrive.