Unfortunately, this is a problem I run into a lot with a lot of lighter-colored hair, but mostly with super light pink, white and blonde hair. Even on non-Hasbo dolls (I'm lookin' at you, Barbie and blonde mermaid dolls :/ ). It's stubborn and doesn't want to un-kink for some reason. If I were to guess, it's due to the hair being dry or broken from the inside-out, so conventional methods of fabric softener or conditioning masks are not going to saturate the hair as much as what we would like. It's a non-porous plastic, after all, so conditioning and softening is only surface-deep.
And, for the love of all that is pony, you don't want to trim the frizz off because it adds weight- removing them can allow broken strands to kink and frizz upwards, so the hair would have been cut for nothing.
Another thing to remember is that flat irons and curling irons are not a tool that can be applied to any sort of doll hair for a constant period of time, and you have to be mindful of kinks if you are not sectioning properly. While it may be easier to some extent, the effectiveness on any degree of frizzed hair is variable because it will eventually evaporate any sort of moisture and may affect the texture in the opposite direction than intended. The less fizzy the hair the better the results- the tools are not meant to be used for an extended period of time in one spot on plastic and vinyl.
So, while backing away from heat tools, there is only one method that has "worked" where it's tamed it a bit for me. But, the likelihood of it being perfectly-perfect is slim. To my logic, boil-perms are used to curl doll hair without using a curling iron, so why not use it the opposite way?
For this, I use heavier reusable plastic straws that I cut in quarters or thirds. Specifically, I use Tervis brand straws because they are not going to collapse under boiling heat (and what I had available the first time I gave it a whirl), and do smaller sections across a longer surface as flat as possible so the strands are not too much on top of each other. I don't start at the very ends, but more 3/4s or 1/2 way from where the frizzing starts. You will need the aid of a spatula to help keep the hair fairly taught, as straight and downward as possible in the water without pulling.
I use bobby pins as a weight, so depending on how bad the hair is, you may need to add more bobbys (flat side on the outside) to the straw, and you need a taller pot to boil water. The ends themselves will be tended to later- you would be starting on the hair well above the ends to straighten them out downward to the tips just in case the strands are broken a bit higher than what we can tell with the naked eye.
I dip the ends well passed the kinks in pure water that was brought to a rolling boil on the stove, and hold it there for a couple minutes (or longer) in the constant heat. The pot should be a 3 or 4 quart size because you want the hair to hang freely in the water and do not allow it to touch the bottom to prevent slack. I remove the pony, let the hair cool, un-pin to comb the strands back over the straw and move downward if it appears that the strands above the straw are starting to lay flatter and have a flatter texture.
There may be times you need to start the straw a bit higher than expected and repeat the process multiple times down the locks, but do not twist the hair around the straw as if you are trying to curl- it's purely to straighten- the goal is to restructure the strands from the "okay" part of the hair down. After the boiling I do the conditioning or softener to restore the outside sheen and prevent tangles.
Sometimes it works really well. Other times it just makes it more tolerable, and I proceed with the full-on curl with everything to hide the ends of the hair that can't be saved. The texture may not be fully restored to like-new condition, but it won't look as if the pony stuck a fork in a light socket and called it a day.
Some have done a finish with leave-in organic dog/horse hair conditioning sprays because they are less acidic and won't "dry" or perfume the hair as much as what human shampoos and softeners have the potential to do, as some sources have said. I have not tried this, but I may as well pass on this information. I personally am not a fan of things being labeled "organic" because "organic" doesn't necessarily mean that it's "good", so this is not a route that I will personally take. Though, I won't poo-poo the method because it apparently has its good points.