How can I secure ownership of a significant project despite my mid-level experience?

You are taking a great career step forward by being proactive and thinking about ways to provide value to your company.

I would start to talk to other people that are within the same domain of the project that you are thinking about working on. This will help in two ways:

1. You can validate whether the problem you are trying to solve is truly a project with wide scope.
2. You will get to know people that can vouch for you in the future. If they see that you've done the due diligence and thought hard about the problem, they will help to enable you to succeed.

I wouldn't worry too much about other people trying to poach your project. People will probably have their own projects scoped out for the next few months, and they're probably too busy to try to steal your project. If you end up being the first to have the conversations, publish a detailed design doc, and set up meetings with different stakeholders, it's really unlikely that someone would try to lead the project since you are already taking the lead on it.

I would also speak your manager about this project. They would probably love to support you on it because if you are able to execute it well, it makes them look good, too. You can probably talk about the resources needed with your manager, too.

You have an idea to help with a company pain point, that is a great!

First do not stress about ideas. Good jobs come with numerous hard problems to solve.
In general the execution matters way more than an idea.
If the company decides to assign the lead to someone else you can still help. It might even be a perfect mentorship opportunity for you.

Think about the following. If you were yo have an important surgery would you prefer an experienced doctor who has done this many times or a newbie that is just learning?

Regardless of the project getting funded or who will lead it I would focus on developing your senior engineer skills. This will require you to come up with designs and get them reviewed.
Senior engineers rarely work in isolation so it is your opportunity to write a design document and present it to a larger audience. The more effort you put into your designs the higher the chance you will be working on that project in some capacity.
One way or another you need to get your ideas into the open and be able to address any feedback. This is an important skill for a senior engineer.

Good luck!

How about you share this project idea, someone else leads it, and the problem gets solved making the world and company a better place? Let's say you don't get any credit for it at all (very unlikely), you still get the experience of coming up with an idea, validating it, marketing it and seeing it go from ideation to deployment. You might be able to follow along with the leader leading it, or take part in some small way in it. But, ultimately the customers would be happier.

Prioritize on adding value, it's okay if you don't get credit. Recognition will follow, if not now, maybe later. There are several more ideas in the place where this one came from, just waiting to be unleashed.

Like others have mentioned, don't worry about someone taking your idea and more importantly, letting your title limit yourself.

You want to enlist others for this project vs. working on it in isolation. Your leadership team is looking for someone who others are willing to rally around. That's actually one of the most important criteria to meet before putting someone into a leadership position in the first place (way more important than functional / technical proficiency).

To change gears a bit, if I were you, I'd focus way more on the PROCESS you took to find this opportunity in the first place and replicate it to find even more opportunities.

If you can recognize the characteristics of a high-impact opportunity, you've solved for at least 30% of the engineering leadership puzzle, which will pay out way more than any credit your receive for leading a single project.

You'll operate at a much higher level after transitioning from a scarcity mindset to an abundance mindset.

To make this transition, the key is having the skillset to find more opportunities any time you want, so the one you have doesn't feel make or break.