There's probably a way to get Arc to do this more automatically, but here's how I did it:
1.) Union your polygons with the polygons as the only input - there will be multiple copies of each overlapping polygon
2.) Run Find Identical tool on unioned feature class to assign a unique number to each set of identical polygons - this is found in a new field called FEAT_SEQ
3.) Join this table back to the unioned table based on ObjectID from unioned feature class and IN_FID from find identical table
4.) Run Frequency tool on the find identical table to count the rows with the same FEAT_SEQ value
5.) Join the frequency table to the unioned table based on FEAT_SEQ
After the Union, I made a new field in the Union output called 'DissolveID' and calculated it as Round(Shape_Area, 4). I then ran my Union polygons through Dissolve using 'DissolveID' as the dissolve field and my unique Id as a COUNT and the value as a SUM.