You mentioned that it is a king size batt, but what brand of batting is it? If it really is an all-cotton batting, then it's fine to iron it as it will hold up similar to cotton clothing. It will compress, but that actually makes it easier to quilt. In the first wash, that compressed cotton will spring back to its original loft.
My concern is it may not be 100% cotton. Polyester can melt under a hot iron, so I do not recommend ironing a polyester batt. 80/20 batts (80% cotton/20% polyester) can be ironed, but with care (and not too hot an iron).
Assuming it's 100% cotton so you can iron it, I would have a spray mister handy to give it a little steam as it is pressed.
The problem with soaking and then placing a king size batt in a home dryer is that the dryer isn't big enough to dry the batt without wrinkles. A laundromat dryer would be fine.
There are multiple ways to get wrinkles out of batting. There is no one "right" way. The right way is the one that works!
Some wrinkling is actually okay. I have quilted at times with some wrinkles in a batting and the quilts came out fine. These weren't huge, deep wrinkles; just not perfectly unwrinkled batts. The first washing of a quilt generally takes care of any wrinkles in the batting.