So much shotgun brood seems highly improbable in an open mating queen  
rearing system unless the breeder really limited not just his queen  
sources but drone sources to a few highly related queens and is very  
isolated.  To get to 50% shotgun brood with multiple matings all  
drones that contributed to the insemination of the queen have to carry  
either one of the two alleles present in the queen.

Other factors? Inviable sperm would produce a different pattern: a mix  
of worker and drone brood in worker cells.  It sounds like your ruled  
out early death of larvae due to the common diseases.

Two groups may be interested in running some samples to help sort out  
the situation (but none can work through the sex allele issue as far  
as I know):

1) The people doing diagnoses of diseases at the ARS Beltsville lab.   
They could check for the routine things and beyond and could suggest  
what samples to send their way.

2) Dave Tarpy's lab at NC state has a funded program to study queen  
quality, but they would need some of your queens.

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