- What do your girls have downstairs?
- How much of the time do they have free access to forage (grass or hay)?
- When did you last weigh them and how have they changed?
- How often do you check each pig for injuries or bodily changes?
This is a bit brief as I need to get out on the school run, but based on what you've said I'm thinking that IF no-one's actually getting hurt, there's a CHANCE that much of the bickering is rooted in your girls competing over resources - both space and food.
Whilst the run is large, it looks to be basically a bare dirt run. What's down there for them, other than space? What temperature is the ground? The air? How breezy is it? Is it sheltered if it rains? Do you tend to get predators creeping around? If they find it a bit chilly and/or damp and/or they feel unsafe, they'll be disinclined to use the space there - which leaves 5 pigs sharing 1m^2, which is a bit cramped.
There doesn't seem to be much grass growing, and I think you've said you're trying to get hold of some hay but they haven't had it as you have allergy issues? That makes it tricky, but Guinea Pigs NEED access to forage 24/7. They should be eating hay or grass almost constantly - its essential for their digestion, and also for making sure their teeth stay in good shape.
Space is obviously a tricky one to solve, but giving them free access to forage might calm things down. Please don't add another pig - if I'm anywhere near right on the cause of the problems adding another pig to compete will only make things worse!
Have you had a trawl through the information guides on here? They should help you try and work out if what you're seeing is just generalised dominance jostling or if there's actually a problem that will need more drastic measures (separation) to resolve. If you want to keep them together then I'd try giving them non-stop hay and see what happens. Good luck