There are not a lot of options for you in a scenario like that!
When I still worked at my old company, we worked on Dead Island 2 with Unreal 4 and it was supposed to open world as well. Sadly, Unreal 4 doesnt support open world lighting techniques atm so you can either hack something together that kind of works, or you implement your own solution (which is what we did in the end)
First of all…some basics!^^ Why do you get blue light inside your interiors?
The skylight supports static, stationary and movable. Static and stationary…everything you see there is just a preview UNTIL you bake the lighting! This is the reason why your interiors are blue! Unreal doesnt know about those walls blocking the light from the sky if the data hasnt been baked. If you use a static or stationary skylight, the blue bright light will go away inside the buildings as soon as you bake. However, there are 2 things that will be very hard for you in that case: 1. you need working lightmaps, otherwise everything will look super broken! 2. Lightmaps is the least favorable approach for doing open world games since you will run into the situation where you cant bake all that data anymore or run into memory troubles with streaming TONS of lightmaps etc. This is actually the reason why almost every open world game uses dynamic lighting and then implements very specific custom solutions to not get bright blue interiors^^
Something you can consider to solve your problem: Use a movable skylight and enable Distance Fields! This way, you get occlusion data for dynamic lighting based on distance fields and you can use distance field ambient occlusion to make interiors look dark because they can dynamically occlude the skylight.
There is a big downside to that though: You will not get any GI! Thats the reason why you would have to invest in custom solutions. One thing you could try is to enable LPVs in Unreal and combine them with distance field lighting. LPVs can make GI but only for the sun…no local lights are supported. You also might end up getting light bleeding inside buildings because of the technical limitations of LPVs (mainly low rez)
What we did for Dead Island 2 was to use distance fields for shadowing and we implemented our own irradiance probe volume approach for localized GI for the sun and local lights. So we basically stored local color bounces from lighting inside probes that we placed in the level and those were then shadowed by distance field AO. However, that needs some programming work to be done.
I think trying to combine distance fields with the LPV could be a good start for you…however, I am not a big fan of the LPVs since the tech is not supported anymore and the code is old and hacky and the performance also isnt too great.
But again…fully dynamic open world stuff is basically exactly the thing UE4 is the weakest engine right now, so you might never reach a result that you are completely happy with without having a graphics programmer to provide project specific solutions.
EDIT: I just had a look at that battlegrounds game, and I am pretty sure that they use LPVs. Of course not a 100% sure, but it looks extremely similar to what we did on Dead Island 2 before we got the custom probes. I also noticed while watching a lets play, that all the buildings are perfectly aligned to the grid! This is why you dont see the typical LPV limitations because the grid of the LPV becomes increasingly more noticeable as soon as stuff is not on the grid anymore. But I was able to spot quite a few artefacts that are very common to LPVs and thus I would say they use those and shadow the skylight with the LPV data. Also, you can see that shadows far away are brighter than we the player gets closer. This is something that happens with LPVs too since they only cover a certain distance away from the camera. So after, lets say 500 units, there will be no effect of the LPV anymore, meaning shadows dont get the additional sky occlusion anymore and look brighter. Hope this helps!