I agree, that small coop is plenty big enough for the hen and 3 chicks. At 4 weeks of age the chicks should have no problems going into that closed section whether they walk the ramp or fly up and down.
That run should be plenty big enough for 18 chickens. I assume you have that small coop sitting inside the main run. That main coop is not that big for 18 chickens but a lot of that depends on your climate and how predator proof you consider your run. If they are locked in that coop only for very long due to weather or you just leaving them locked up in there I would expect problems and a very challenging integration with those 3 chicks. If your climate and management techniques allow them outside during waking periods you can be OK.
Penny seemed to think she could bully Bertha and her babies out of the crate because there was food in it, and Penny is pretty greedy when it comes to food. A fight ensued over the old living area and the food, and I had to break up the fight and lock Bertha and the chicks in the crate.
This complicates matters. I think Penny was after the food and Bertha did not want her near the babies. If it were just the hens fighting I'd be tempted to let them go, they should soon sort it out. But my concern would be that the chicks could be injured by the fighting.
So how can you avoid this? I assume the broody and chicks are locked in there overnight for predator protection and the others are locked in the main coop. My first thought is to let the broody and her chicks out first, before you open the main coop and let the others out. Also move the feed dish out of the small coop into the run to encourage the broody and chicks to come out and to keep Penny out of there. The broody and chicks will probably still want to eat when they wake up and Penny may still be her obnoxious food hogging self, but at least it would be in the open with less danger to the chicks. I think they will sort it out fairly soon. I'd consider adding another feeding station far away in the run to give them another eating option.
Also, what do you mean by the hen weaning the chicks? Like how does that work and what are the signs the hen is weaning the chicks?
At some point the hen will determine that the chicks can make it alone in the flock. You don't know when that will be. I've had some do that at 3 weeks, some at almost 3 months. Some hens abandon their chicks and want nothing to do with them essentially overnight and rejoin the main flock. They might peck them to drive them away or just abandon them.
I've had hens continue to sleep with their chicks at night but not take care of them during the day for another week or two. I've had hens stop sleeping with the chicks but take care of them during the day. Most of the time it is an immediate total abandonment but with living animals anything can happen. It is a normal part of chicken development. I have never had a hen wean her chicks before they were ready to make it on their own.
With your very small main coop the chicks may continue to sleep in their small coop for quite a while after being weaned or they may try to join the main flock in the main coop. The broody may move them into the main coop before she weans them or she may abandon them out in the small coop while she moves back to the main coop to rejoin the flock.
Good luck! With your set-up you should be OK. It's just that food crazy hen that is upsetting things.