Can you arrange set days where you take them out? For example you take one out Tuesday and one Wednesday one week then swap days the next week so they don’t argue over which is the better day?
If you do the same activities then you can’t be accused of being unfair as you are doing exactly the same. If they continue to scrap over it then just say you don’t want to spend any time with them, it might be an incentive for them to behave a bit better for your friend.
Are they just fighting for your attention or is this a constant thing when you aren’t present? If they are fighting over you then I’d plan to see your friend without their knowledge. If this is all the time and you are trying to give your friend a break then you have the patience of a saint! I think it’s kind of you to try but I wouldn’t be able to cope with this.
It sounds like your friend needs outside help managing their behaviour, she must be exhausted having to deal with this constantly. Is she punishing the behaviour when they are screaming, yelling and slamming doors?
It sounds like she needs to punish both equally for the bad behaviour and not award it with more attention, they are both going to escalate the tantrums to compete for the attention. If they start scrapping they need separating and putting in different rooms to cool off then your friend only responds to the good behaviour. Does she spend an equal amount of time with them separately or is this not possible? Is their father involved? Could you plan a day out with one child whilst the other has a mummy - daughter day?
If it happens every time you visit then just leave as soon as they start arguing, it’ll teach them that it’s not gaining anything and you don’t have to put up with the tantrums, if your friend isn’t doing anything to stop the behaviour then it’s just chaos you don’t need in your life!