I worked with a few guys at uni to remap the battle of Gettysburg and I sorta came to the realization that his move may have won the battle.
I was criticized for thinking this and I kinda just want to know if the movement actually changed anything in the battle
Not in the least. A salient can be hammered from three directions and so can be a weak spot- and his was. Sickles' move opened up and exposed the Union line to an attack by Confederate forces and without quick action to close the gap it could have cost the Union the battle. There's a very good discussion and outline of it here by u/LarryMahnken.
I can recommend Thomas Keneally's American Scoundrel if you're curious to know more of Sickles. The title says it all.....