So as well as the client not being entirely happy with you, you're not even in the same country as them? And there is a colleague available who is in the same country as them who has a good relationship with them?
I'm sorry but it really seems a very sensible business decision to promote them instead of you to this role. From the client's point of view they'd probably consider it a bit bonkers to have an oversight in another country who they don't have confidence in (by the sounds of things) if there is someone they have a good relationship with based in their own country who can do it.
Life isn't fair, and sometimes other candidates have attributes or circumstances which might make promotion or recruitment decisions seem like an 'unfair comparison' - but if you think of it from the client's point of view, and from your bosses' point of view, it is an obvious decision, especially when dealing with a difficult client, to have someone local who gets on with that client being the one dealing with them.
Yes ideally clients shouldn't have a say in internal personnel decisions, but back in the real world, some pragmatism is needed to maintain good relationships with clients, assuming said clients are valuable to the business, which I assume this one is.
I still don't get the 'not rolling over' bit. If you don't get an internal promotion and it goes to someone else instead, it's not a case of 'rolling over' or not - you have to just accept it, as your employer is allowed to make that decision.
Your only options are to either accept it and continue to do your job as well as possible, or you look for another job elsewhere. I don't think either of those is 'rolling over'.
Whether your company will ever award you a promotion, who knows. If another opportunity comes up and in the meantime you've done all you can to prove yourself, either with this client or others, then they might, but no one here can speculate as to whether any opportunities will come available, how influential this particular client is, whether the hypothetical promotion will involve dealing with that client, or any of the other factors involved.
It sounds like there is a bit of a communication problem with your boss, if you're finding out things by 'seeing internal documents' so I suggest you have an open conversation with your boss, ask for clarification on the process for the upcoming promotion opportunity, and take it from there.