I know very little about both settings, actually, for a person that's been playing D&D like forever. But my personal view on the Multiverse is--I imagine!--in line with that of the D&D creative team, which is that every D&D setting (and in my personal headcanon, plenty of planes of existence or dimensions that AREN'T D&D settings) is connected. I'm not sure why your dislike of FR's religions is so intense that you need to ensure that the dimension of Eberron is SEVERED from the dimension of Faerun, like, I feel like for 999/1000 campaigns, simply NOT having the PCs planeswalk to the setting you don't like/don't want to use is more than enough.
Think of of it like star wars & star trek. No amount of handwavium can fit luke skywalker onboard the enterprise, discovery, or defiant without causing massive problems to either what luke skywalker his lightsaber/jedi powers/struggles or destroying the setting those ships exist in to import luke & his baggage. By extension no matter how much you are willing to look the oher way, the
IKS Bortas & crew will never fit in under vader/palpatine/Snoke/etc without violence either to that ship & its crew or the fabric of the Star Wars' reality itself.
Part of the reason eberron fans are quick to bristle at the idea is likely due to a few reasons & you've seen some of that in this thread. Some settings (ie FR) tend to break down to killing gods & other megapowerful individuals after growing ultra powerful yourself while eberron is more of a complex system balanced on a knife's edge with few clear paths of right & wrong where that sort of direct detect evil>smite evil can easily result in the
hero breaking things monumentally. Villians are organizations & sometimes beings with
hard to grasp goals or trivial to understand yet seemingly impossible to realize with just a murder. You wind up with people comparing CR9000+ eleminster's statblock to the more nebulous ones of things in eberron that are expected to grow with the PCs & if you need their statblock things are already going horribly wrong.
Some of the other problems that cause the bristling are the fact that the first of ten (now better worded seven) things you need to know in 3.5/4e eberron books was "1. If it exists in d&d,. then it has a place in Eberron" which would result in people not bothering to finish reading point one or even glancing at points two through ten. Now that's moved to point seven & includes some of the other "missing" points as
"D&D with a Twist. Every race, monster, spell, and magic item in the Player's Handbook, Dungeon Master's Guide, and Monster Manual has a place somewhere in Eberron, but it might not be the place you expect. " which is much better phrasing that doesn't simply open the door to wholesale import of another setting & all of its conflicting lore on top of eberron like a motor oil glazed sugar cookie(bad for you, bad for your engine).
Also in 4e there was lots of pretty problematic stuff from other settings complete with supporting lore actually published in eberron sourcebooks in ways that conflicted with the setting itself so you get people quick to protest if it even looks like it might cause a smell similar to that kind of thing.