Yes! You can use the same actual business entity/LLC for registering as a Publisher with both ASCAP and BMI, however your actual publisher affiliation names with each society have to be different.   

For example, if I wanted to setup my publishing entity, Smulders Publishing LLC, I would name it as follows, since I cannot use Smulders Publishing LLC for both:

ASCAP - Smulders A Publishing LLC
BMI - Smulders B Publishing LLC 

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