7/8 hole through boxed joists
You say, “less than 4’ wide.” Therefore, only 2 joists At 16” oc could be supported by the header. 

Generally, short spans are limited by shear, not bending. Your “box header” will support about 800 - 1200 lbs. per linear foot, depending on the species, grade, etc. of the wood. Can you fill in some info?

You say, “two story home.”  What rests on that bearing wall and header? (1 floor, 2 floors, 2 floors and a roof?) What is the span of the joists that rest on the header?