I am attempting to intersect two feature classes, one of which has many overlapping polygons. I don't want to calculate the intersection of overlapping features within the same FC, but instead only the intersection of features *between* the two FCs.

My problem is similar to [Intersecting overlapping polygons individually in ArcGIS Desktop](https://gis.stackexchange.com/questions/87940), but the solution proposed there will not work in my case (I'm not trying to calculate zonal statistics.

I've thought of two potential solutions but they each have their drawbacks. First was to simply dissolve the output of the intersect based on the FID field of the FC with overlapping polys. However, this create problems with preserving the format of the attribute table which I need for future calculations

My other idea was to export each individual feature to a temporary FC, then do the intersect and add the intersected features to a new output FC. I think, however, that this would be very slow (i'm doing this over 10,000s of features w/ arcpy).

Is there another way to do this?

(EDIT) Example:
The polygons outlined in red are from the feature class w/overlaps.
[![enter image description here](https://i.stack.imgur.com/2fYIr.png)](https://i.stack.imgur.com/2fYIr.png)

When I do an intersect w/ the purple polygons, the result has a 'sliver' where the two red polys overlapped (highlighted below). This is what I DO NOT want. 

[![enter image description here](https://i.stack.imgur.com/aNRL1.png)](https://i.stack.imgur.com/aNRL1.png)

In this example, the only intersection I really care about is the one highlighted below: 

[![enter image description here](https://i.stack.imgur.com/c7whg.png)](https://i.stack.imgur.com/c7whg.png)