How to approach skip manager with specific task requirement?

Similar to Lee, I'm also a bit confused by the question - Can you add more context on what "end to end ownership" looks like to you?

How do I approach him to ask that I want to work on a task with end-to-end ownership?

Is the idea that you don't own this current project, because it's owned by the sister team? I have seen this sort of "loaning" situation happen a lot before, especially in bigger companies like VMWare. It can often feel less rewarding to engineers as they want to work on their own thing.

If this is the case for you, I recommend talking to your manager about how much credit you're getting for your efforts. Generally engineers, especially senior engineers, are rewarded for working across teams, but this will obviously vary based on the ownership structure and your contribution.

By end to end ownership I meant I want to take end-end ownership of a feature on this project, basically drive a task from requirements to completion

Ah okay, got it. I think you can more or less take this and ask them. Something like, "I really like this project and want to take on more responsibility. Do you think I can own X part of it and own it from requirements to completion?" I don't think this is a strange ask, especially since you're a senior engineer.

On top of this, I have 2 more trains of thoughts:

1. You should do this naturally - Something that got hammered into me as I was growing into a tech lead at Meta was "Fill in the gaps." If you want to own more scope end-to-end, you should start identifying problems in projects on your own and proactively solving them. Once you build a reputation as someone who's always willing to do whatever it takes to move the project forward, you'll naturally find large end-to-end ownership naturally falling on your plate. For a very good in-depth example of this, check out my case study where I talk about becoming a tech lead again in just 1 month after onboarding at Robinhood.
2. Be transparent about your aspirations in general - Promotion and growth in general should be an ongoing, transparent dialog with your manager, primarily done through your 1 on 1 meetings. If you haven't already, try working with your manager to create a growth expectations plan. Doing this will make asks like the one in the question less awkward and surprising when you bring it up with your manager.

This isn’t completely clear. You said you’re working on a project with a sister team already. Aren’t you already working across teams?

Why not start with your manager, even if they don’t run the project itself? They may not be able to allocate the specific work, but could determine if this is how they want to allocate your time and communicate with the right people.