I had the same need as you to have F-Droid and Aurora store as priv-apps. I now build a custom ROM for my device with this included, which works fine for me, but I understand why you are not keen on the idea.
Before I built my custom ROM, I used NanoDroid to install a variety of things, including F-Droid and Aurora Store. This post gives details of how I used it to install a working version of MicroG in place of /e/OS’s fork. Full documentation available here
If you only want F-Droid and Aurora Store, then you don’t need to flash the whole package, just flash the ’ NanoDroid-fdroid module. I’m pretty sure that if you flash that zip file (via adb sideload) immediately after installing /e/OS, without first booting into /e/OS, then you should get what you want. Definitely worth a try anyway. Once installed, F-Droid will keep them up to date.
Good luck! 