How to escape E4 work as an E5, so I can focus on E6 work?

There's 3 approaches here:

1. Hire an E4 or E3 - I imagine this is much easier said than done given the current economy, but an option if you aren't able to hire externally is to procure a loan from another team. If you can't get an E4 but can get an E3, you can use this opportunity to uplift them through mentorship (every E3 should be excited to get E4 work). However, you got Exceeds in your last review (congrats!), so this doesn't seem to be a huge problem. That being said, this is probably a conversation worth having with your manager to test the waters, and it's in every (good) manager's best interest to empower their E5+ engineers to scale themselves.
2. Expand its scope - A common misconception I see among engineers is that they believe that tasks and projects have an intrinsic "level" of scope, which ends up holding them back (e.g. "This project is E4 scope, so it's not enough for me"). I see everything as having more of a "base" level of scope that is adjustable based on your behavior and creativity. For example, an E4 project can be "downgraded" into E3 scope if you deliver it very poorly, but it can also become an E5 project if you expand its scope. For tactics and examples around how to expand something's scope, I highly recommend this discussion: "How can I come up with big initiatives, especially those at a Staff level?"
3. Get it done faster - When I think of E4 work, I think of something that's more coding-heavy than it is tech lead and XFN heavy. If you really can't get delegate it away or expand its scope, apply the techniques from the "How To Write Better Code Faster" masterclass to minimize your time spent there. At the end of the day, work has to get done, and sometimes there won't be an appropriately leveled person to take it on. As an E5, your goal is to fill in the gaps, so if there's no other solutions, you simply have to roll up your sleeves and do the work (I did this all the time back at Instagram when my team was understaffed). Since you're E5, base-level E4 work should be quite easy for you and is a nice exercise to build up your muscle memory.

Lastly, I hope this thread is helpful as well: "How does one effectively handle pressure especially when the stakes are high?"

If there are two people that can do this work, you and an E6, and the work that needs to be done is “lower level”… it has to be done. You can try to decompose it so you can both knock it out in a week or two, you can have a “get it done” rotation by project or month or something… but the bottom line is they have seniority. You should be shadowing them on the work they are doing, contributing where you can, etc. but sort of have to accept they are better suited to take on that work now. There’s times that simply require delivery, and times you can stretch and grow. With two iOS engineers, though, unless y’all move to a shared code setup that allows other devs to contribute, you cross train, etc there probably isn’t ever going to be a time you’re not building workaday features.

It sounds like hiring an E3 or E4 isn’t an option, not sure about cross training, but basically this isn’t a team to grow on with the current composition. Maybe you can build something that allows no-code/low code feature addition to accelerate that stuff, but that seems lofty.