Your children's room xmarker should be in the room, not the void. The game uses it to tell the kids where to go. If it's in the void, they can't get to it, so they don't know where their room is. It should have the appropriate location reference type. I can't see it in the image so don't know if you did this. When you refer to a keyword, I assume you mean Loc Ref Type. Loc Ref Types and Keywords are 2 different things.
Using two won't help at all. The game will look for one in the location and will not know which one of the 2 to pick, the way the adoption quest is set up. It may choose one, it may not assign any. I don't know without looking carefully at the scripts. The issue is elsewhere in how you've set up the cell. You most definitely only need one and only one.
Roombounds are nothing to do with it. They are used for optimization purposes and can be used to assign things like different lighting templates to different areas of the same cell. They don't need to come into this at all.
I strongly suspect that the entire cell is not set up correctly for spouses, housecarls and adopted children.
The guide provided below tells you how to set all this up for 'Hearthfire multiple adoptions'. Follow it and you will be OK. I've used it and it works fine, as it does for HMA. Follow it exactly. The maker of Hearthfire Multiple Adoptions went through all the relevant quests and scripts and provided this guide so we don't have to! Sweet.
With the marker in the correct place and the two beds with the right loc reference types, it should just all work, providing you have all the other stuff in the guide in place. If even one of the things that the adoption script looks for is missing or lacking the correct location reference type or keyword, it will probably fail entirely, so no adopted children will ever show up at Breezehome.
As for the PS4 comment, I never intend anything for PS4 because of its restrictions. Things like this that relate to basic game mechanics should apply equally to both PC and PS4. Others may advise more on this. For now, let's get this problem solved!
Edit: I'm a little concerned about the rest of it, since it looks nothing like Breezehome. Where's Lydia, for instance? She should be standing close to the top of the stairs, unless you've just hit '1' twice to hide her. One of the kids' rooms looks like Lydia's room. Where's her bed? You haven't got markers enabled, so I can't see if the rest of it looks fine. I'm also not seeing any signs of a whole raft of statics and activators that should have been dragged down below the house, along with the vanilla navmesh, unless they are out of shot or hidden.
If you've deleted vanilla objects*, or moved Lydia to another location, you're in big trouble! And I mean delete your esp and start again kind of trouble! If you've deleted vanilla navmesh, you may get out of this, with some effort in SSEedit. I can guide you on this. I'm not saying you have, but I suspect it. Let me know, so we can rule it out.
* statics, navmesh, activators, you name it, you delete it and woe betide you! Chaos ensues.
Anyway, that's my answer to your original problem.