I'm obviously not a networking person. As far as i know you need a router between modem and switch. Even if it is possible via a fancy switch and actual IT expertise... you said "backfeed using the same cable". I think this requirement slams the brakes on your plan.

I cant tell exactly what you are doing but it sounds like you want to use the EERO as a router...but you don't want the eero where the modem and switch are because you need the eero in a specific room...where you only have 1 Cat5e.

If i guessed correctly you should just buy another eero for the "equipment closet" or other 3rd party router.