As others have said, unfortunately hair can get very frizzy if it is brushed when dry or when over brushed with a plastic brush/comb. So you do need to get it wet to comb it and ideally use a metal comb. However, because you get the hair wet does not mean you have to get the plush wet. It just means you need to be clever and careful with how you wash it. Saran wrapping the pony in order to protect her body and leaving just her hair uncovered is a useful starting point.
I would wash and condition the hair to start with - you can do 3/4 of the length to start with and see if it works. Soak the hair over night and also wrapped up in saran wrap.
G1 plush hair is such that you can use a regular flat iron on it. I am not certain on the BAB plush. You could remove a handful of threads of hair from her tail however and test a flat iron on it. If the hair is wet and you use the lowest heat possible you should be fine. As to avoid singing the plush - just dont put the flat iron near it and again perhaps only flat iron an inch from the root of the hair.