@LeadFaceJake
Can't offer you a solution with exactly what you specified, but if you are willing to do some sacrifices in terms of WiFi speed and money, I have a solution for your consideration.
I happened to have old PoE injector Linksys WAPPOE12, which appears to be what you may need, and I checked what it has. Its power adapter DC barrel plugs perfectly in my Deco M5, assuming M4 and E4 have same barrel plug size - it should work.
This PoE injector does not support gigabit, only 100Mbps. Which will reduce WiFi speeds to that, obviously. Also, according to feedback from users, its splitter will not auto negotiate if device has gigabit port. Injector part works with gigabit networks just fine, I used it at my house and can confirm that.
Also, I can't find exact specs for DC output, strangely enough, but from the list of compatible Linksys router models it appears WAPPOE12 delivers 12V=1A.
Deco E4 is listed at 12V=1A. Deco E4 has 100Mbps port, and that is exactly what you'll need for that solution.
So, if you go this way you'll need Linksys WAPPOE12 and one Deco E4. Also, make sure you confirmed DC barrel plug size is compatible with E4, which it should be if E4 is same as M5 in that respect.
The back of the box with Linksys WAPPOE12.