I think the F-82 had a bigger spinner. The actual shape of the fuselage is NOT identical to a P-51 of any sort.
Unfortunately, I have not seen the only F-82 operating these days up close and personal, but the spinner LOOKS bigger. It may be bigger, or it may be the same size and the different fuselage lines are fooling us. It is basically a P-51H fuselage with a 67-inch fuselage plug aft of the cockpit area. So, it is likely to be the same spinner and prop, logically. Looks different because the fuselage looks different.