As far as I can tell, it is a pecking order situation, but my Austra also has a definite dislike with sharing food, so she tends to be more cranky with my Red and Sex-link.
Do you feed your chickens only once or twice per day? Do they eat all the feed when fed?
I have read that if you only feed a small amount to your birds once or twice a day, they will learn to fight for their food and gorge themselves when the food is offered. One way to "correct" this situation is to have feed available 24/7. I have my commercial feed available in a 5 gallon bucket feeder that hangs in the coop 24/7. I never let it run out. My birds have never fought over their commercial feed as they have learned that there is food available all the time. They are free to eat a little bit whenever they want so there is no fighting at the feeder.
I want to prevent an issue before it is an issue. so... back to my original question, revised: when integrating pullets to an aging flock, should I get more than one?
I would get more than one, especially if you intend to eventually replace your ageing flock.
Does size/age matter?
Yes, size matters, not age so much. The pullets will have to be big enough to defend themselves or they risk being killed by the adult hens.
(I am not concerned about coop and run size.)
The first thing I would be concerned about is coop size. You mentioned you have a small coop. Small coops lead to bad behavior for your flock. Introducing more birds to your small coop many increase the bad behavior you have mentioned. I understand you might not be able to "upgrade" your coop situation, but if you have a small coop, maybe you need to think about reducing your current flock before getting any new pullets.
There are some local farm stores that sell prefab chicken coops that are really only big enough for 3, maybe 4 hens, but they advertise the coop will hold 8 hens! Again, most people here would recommend a minimum of 4 square feet per bird in the coop to keep the peace. Double that recommendation if you live in a snow bound state for the winter when the hens will spending most of their time in the coop.
Also, as to the chicken run, I have read that adding "run away" shelters for smaller pullets can be effective to help them get away from a larger bully hen. Provide them places that they can run under - and through - making sure they cannot get trapped inside the shelter.