Removing overlapping polygon parts in sequential order using ArcMap

I've done something like this in the past.
I Selected the top priority Polygons, Erased everything overlapping.
This output will be the input to the next step.
Select Priority #2 Polygons, Erase everything overlapping.
continue...

Since you only have 3 Criteria in the same Field, these steps might work in Model Builder.

Create separate FeatureClasses, SplitByAttributes.
You now have 3 feature classes FC1,FC2,FC3 (name your data to fit your needs)

FC1 is not edited because it is the top priority.

Erase FC2 by FC1 producing FC2_erase
Merge FC1,FC2 producing FC12_merge
Erase FC3 by FC12_merge producing FC3_erase
Merge FC1,FC2_erase,FC3_erase producing Final_FC

We resolved the issue using the Feature to polygon tool, to get the shapefile without the overlaps. The result was used as target in the Spatial Join tool. The attributes were joined based on spatial location (intersect/search radius: - 10m) and on the field containing the sequence (Field Map of Join Features -> Right click on field containing the sequence -> minimum).