I wouldn’t think knitting the hat inside out would be a problem, until perhaps the decrease rounds on the top ( if you are knitting it from the bottom up.) I also don’t think you would mess anything up by pulling it through the center --I would put point protectors on the ends of the needles to prevent dropping stitches, but otherwise you should be okay. With circular needles, when you want the outside to be on the outside, you want to knit with the points of the needles in the front, closest to your body. What you are likely doing is knitting with the points of the needles away from you and the body of the knitting (on the cable) between your body and the needles. What you are doing isn’t wrong–in fact when doing a complicated fair isle or stranded knit you may prefer to knit inside out so that you can more easily see that the colors are carried loosely along the non public side of the work.
The bottom line is that you are doing just fine–no need for asprin!