Turns out turning off n-key rollover (NKRO) on my Ducky keyboard finally allowed me to get into the BIOS setup on my DE3815TYKHE. With NKRO on, F2 (and F7, but not F10) was being ignored during the boot sequence.
I can find no mention of this particular issue on the forum so maybe this post will save someone a few hours of troubleshooting.