Howdy dbragg
At 4 months of age she should be OK but being on her own will be tough as she will not have anyone to buddy up with.
Where did you get her from? If outside sources, have you put her through a quarantine period? She may have something nasty that you do not want her to share with your existing flock.
I have not integrated one chicken but I use the ‘look but not touch’ method of integration using wire or a crate etc so that they can get used to each other without the threat of injury. I keep them separated in this way for at least a week.
I let the existing flock out for a free range while the newbies are locked in the coop and run. This gives the newbies the opportunity to explore and feel comfortable in what will be their new home without the threat of being picked on and also learn where all the good hiding spots are.
Same with free range, but reversed; I let the newbies wander around the garden while the existing flock are locked in. Granted, the existing flock have a good whinge about this

but again, it gives the newbies chance to explore, get settled and find all the good hiding spots without being picked on.
I find that free ranging is a good opportunity to let everyone get to know each other .. space is a great tool when integrating.
There will probably still be bickering while the pecking order reestablishes itself when they are integrated, but it should not be quite as brutal.
Also, when they are integrated into the run and coop, for a couple of months I ensure that there is more than one source of water and food, just in case the newbies are being bullied away from either.
Hope that helps