The screen rolling tool you have should be fine to use.
Andersen window screen metal spline.
With the screening tool place the wheel towards the outside of the spline and start rolling it in that way if you slip the wheel of the tool will jump on the outside of the screen instead on top of your new screening.
First roll in the wire on one side of the screen frame that will make an indent so you can place the spline in position.
The spline is white aluminum square stock.
We recommend our 59 250 screen spline for andersen screens but we find that the aluminum wire holds in better than fiberglass in these older frames.
I asked at a home center and they suggested replacing with foam spline but the white square spline shows from the outside.
I can not think of a way to remove it without twisting and or bending it too much to reuse.