I'm working with two political boundary shapefiles from Brazil. One is a municipality shapefile, and the other is the broader micro region, which contains all the municipalities. I'm trying to aggregate an attribute from the municipal file to the micro region one, with the ultimate objective to get the sum of that attribute for all the municipalities that are contained within a specific micro region.
I've been using the spatial join tool, with "Join-One-to-One" and the attribute having the rule set to "Sum". However, when I cross check the results in the attribute table for the new micro region file (the newly joined output) with the labeled data from the underlying municipality file, it's clear the numbers are incorrect. It seems like "Intersect" is the most sensible option for describing the "Match Option" but I might be mistaken.
Any ideas as to what I might be missing?
In the screenshot, the N1994_FEDE value shows 3, but in the labeled map it is displaying as 2, which is the correct value. The first shows that there are only two values in the micro region; the second shows with annotations that the attribute table of the newly joined file is not displaying the correct sum, only adding the 1 value. Using One-to-One and Contains center within as the criteria. 
Update below
In the following two screenshots I show what is happening when I try to use the suggestion of @FelixIP, 



