There are a lot of misconceptions in your question. Some of them are:
Visibility comes from success on visible tasks
Perhaps if the people you work with are all idiots, but otherwise not. Someone decided this long and important task needed to be done. Someone will notice when you have done it and done it well. Make sure you report in a visible and enthusiastic manner. Not "nothing really, just slogging away on the mammoth thing, you know", but "I'm over halfway done on the important thing, and can see finishing it this sprint. I'm looking forward to seeing [benefit this will bring.]"
Your peers are your competition when it comes to visibility
Your peers and immediate supervisors, as well as the people your work actually helps, are the source of your visibility. These are the people who talk about you, who remember what you did and how you did it, and who care about the projects you're part of. Do they see you as a generous and helpful person, who is smart and hard working, and wants the best for the people who use your product? Or as a selfish grasper always concerned about the impact of every single sprint on your "brand" and the "arc of your career"?
One or two sprints - a month or so - can make or break you
Visibility is a slow-and-steady game. You keep showing up, you keep doing great work, you keep caring and helping and contributing. You keep saying the right things in meetings and taking on the tasks that some people don't want. You keep asking how you can make things even better. People begin to understand that you're a big-picture, can-do, smart person, not a "what's in it for me" smart person.
Feeling helpless and angry is an appropriate response to a plan not going as you wanted
Stop thinking "how can I get a more visible task?" and start thinking "how can I make my value more visible?" Stop feeling helpless and starting helping. Stop feeling angry and ask yourself what you're going to do about it. I know, easier said than done, but stewing and sulking will not help in the long run. Understanding (and sharing) the value of this mammoth task will. So will asking if it can be someone else's turn the next time one of these comes around - but not for visibility reasons. Ask because you've done your turn and you'd like a quicker task with more client connection for the next sprint.