I need it to be as straight as possible
Straighten it with your fingers as well as you can, then take a couple of short 2 X 4s (pieces of wood) and roll the straightened clip between them. In the old days, when nails were handmade and quite expensive, this is how they’d straighten used nails.
You can also use this method to straighten wire coat hangers, if you want to use them for crafts.
First open it as gently as possible undoing as much of the bend gradually, so it doesn’t form a harder double hump at each bend. Then use your fingers to apply broad pressure at the center of any remaining curves to make them longer – in some cases this will introduce a much longer curve in the opposite direction, so work that out. Then get a hard flat surface – ideally a steel plate or a flat on a vise – and a hammer and roll the wire so that two points are touching and there is a rise in between and tap on the rise to just barely drive it down to the flat. Roll the wire and repeat until you can roll the wire with no unevenness.
Once you have done a few, you will realize exactly how much force and how much bend is needed at each step and instead of taking 10-15 moves to straighten, you will get it in 5-6 bigger smoother, just enough moves