The Why:
Let's paint a painful specific picture; you're a vampire, but you haven't let that get in the way of starting a family. You're also married to a vampire (not that they had much choice in the matter), and you've even decided to show some kindness to some poor unfortunate souls by adopting a child or two (let's be real here, they're an all you can eat buffet, but hey, at least they're not on the streets any more).
But every family needs a canine companion, and most dogs overreact to the whole... undead thing. If only there were dogs out there that understood you...

You get the point.

One of my longest playthroughs was with a vampire, and I had kids, and the thought crossed my mind of a family death hound. I thought it was funny, but that was back on a console years ago, so I wasn't able to do it. But the idea has been nagging at me relentlessly while making my load order, and I realised that no one had done a simple version of this for SE yet, so here it is.

The How:
I edited two FormID lists: "BYOHRelationshipAdoption_PetAllowedRacesList" & "BYOHRelationshipAdoption_PetDogsList" using xEdit. These two bad boys control what the adopted children see as "dogs". By default, they only react to "DogCompanionRace" which means Meeko, Vigilance, and The Stray. All I did was add the Death Hound's companion race, and now the children can think they're as cute as I do.

Huh, now that I think about it, I don't know if the children would adopt the Dawngaurd huskies since they have a unique race. Something to look into later.

The Compatibility:
Should be compatible with everything, as long as those things don't edit the two FormIDs mentioned above. Mods that change dog companion stats should be fine. Also, any custom Death Hound followers that have the default race should also be adoptable, which could possibly cause a conflict if that modded Death Hound companion was a part of some heavily scripted and specific questline... but that's an edge case.

The Update:
So, when I tested the huskies, the children didn't react to them at all. That might just be an issue on my end, but the changes to the main mod file fixed to for me. I added it to the main file because, why should people have to install two mods to fix the same problem with two sets of companions?

There is also now an option file - It covers everything else, along with the Nix Hound and Bone Wolf pets from the Creation Club. It's optional because it's untested. I don't have the CCs that these pets are from, but I do have access to the companion races in the creation kit. Assuming, of course, that the pets even use the companion races. I don't know, maybe those were redundant? 
Anyways, the optional file *should* work, but it could potentially cause some sort of issue with the two pets, so use that version with caution.