How can I position myself best for success given a shortage of senior engineers on our team?

Getting feedback from seniors doesn’t seem the same as your project being led by them. They will be overseeing everything, even if it’s led by an E4.

Get a mentor outside of the team. I would say a tenured E4, or newish E5. Tenured E5s might be able to focus some time for you, but they should be getting tenured E4s to E5. Ask if one of the E5s on your team can be a sounding board or ramp up buddy or whatever, that casts it differently than mentorship. Say you will rely on the project lead, but also want a focused outside perspective.

Getting these different sources of feedback will be helpful, versus relying on a single project lead to be your only source (who may also have a skewed opinion because they are so deep in the same project).

I want to note that 30% senior, 50% mid, 20% entry level is a fairly senior leaning team. Your classification of “only” 30% seniors seems a bit skewed. Many teams are lucky to have one or two seniors.

Level matters, but I think you may be focusing on level a bit too much. Anybody can get good feedback from anybody else, regardless of level, and this is especially true as an E3.

...one of the points made was to find a way to get feedback from senior engineers at the end of the half who will bolster your PSC packet versus a mid-level engineer who might not hold as much weight...

In general, an E5's feedback will have more weight than an E4's come PSC, but there's a lot of nuance here:

1. You still want the E4 feedback - It's good to have a diverse collection of feedback across multiple levels (this is one of the goals of 360 feedback). Since your immediate goal is to go from E3 -> E4, it would be really weird if there wasn't a single E4 deeply vouching for you. One of the easiest ways to prove that you're operating at the next level is to have a bench of folks at that level consider you as a peer on equal footing.
2. E4s are closer to you, which sharpens their feedback - A lot of E5s, especially if they're well on their way to E6, don't really code that much anymore. However, your primary contribution as an E3 is to become extremely proficient at writing high-quality code independently. This makes it much more likely for an E4 to really "get" you and what you're doing. An E5 tech lead will often have very high-level feedback for an E3 that doesn't count for much on PSC, something like "This engineer is easy to work with and gets stuff done on time."
3. E5s usually don't invest much in E3s - The most common mentorship is to go just 1 level down. An E5 gets far more credit on people axis for uplifting E4s than they do for uplifting E3s. Extending this, it makes a lot of sense on both sides for an E4 to uplift E3s. The incentives just align the best when this happens, and the value add is easier as people generally remember what it was like being the previous level (it's much harder remembering 2 levels away).

I'm pretty sure you can go from E3 to E4 without having a single E5 on your PSC feedback packet, especially if those E4s are on the higher end of the band (i.e. they have made sufficient progress to E5). If you really want an E5 on your packet, you don't really need more than 1.

Should I avoid certain projects that are led by E4s?

Definitely not. You should just avoid projects where you think the leader isn't good at leading or you don't get along well with them. This is true regardless of level.

Does this even matter?

Good question as it doesn't really for E3s. E3s aren't held that responsible for impact, which means that if their projects had a bad TL, they can still do great in PSC. Let's say you're on some project doing the back-end portion. You turned in all your code on-time and it's super clean, but the project overall collapses as the TL didn't address XFN dependencies proactively. You wouldn't get dinged at all for this, and you would get rewarded for your excellent code contribution.