When you say an IO-360, do you mean the 180BHP version or the 200BHP version ? As I believe the larger one as fitted to Bulldogs are actually a bit bigger. The G-RVVI that Jerry Parr built has one of those and has a non standard bulged cowling ? Not sure if the PA28 Arrow with both the 180 and 200 are in fact the same engines.
So far as I know same crankcase with different cylinder heads. The other main difference is the 200 HP variant is usually fuel injected.
The 180 HP engines have parallel valve head where as the 200 HP have angle valve heads. The angle valve heads have wider wider head and wider rocker covers and thus require more space under the cowls. They're quite easy to distinguish from the parallel valve versions. The parallel valve head rocker covers are bowl shaped, the angle valve head rocker covers are flat and bigger.
I'd echo Steep Turn's comment, don't cheap out on the engine.