> since the queen is too engorged with eggs to fly.

They can fly!  Some years ago I was re-queening one of my hives in the centre of Stratford and a hive in our Association apiary needed a queen.  I took out the queen from my hive, caged her and put in the new one.  Then went to the Association apiary which was only a couple of hundred yards away - but with two rows of terraced houses, 3 stories high in between.  For some reason that I do not recall, the queen was marked but not clipped, so I decided to clip her.  As she was heavy and in full lay, I did not think that she would fly so worked on the parcel shelf of the hatchback with the hatch open.
Of course, she flew.  I watched here circle round with her abdomen hanging down. Slowly gaining height and then heading off over the houses towards my apiary.  I returned to her original hive and after a short search I found her, re-caged her and started again - this time working in the car with all the doors and windows shut.
This was a two year old queen in full lay; not only could she fly - but she had also remembered the hive from which she took her mating flight two years previously.

>... house bees immediately engorge on honey...

Good point - it is the house bees that engorge - and they do not usually sting!

Best wishes

Peter 
52°14'44.44"N, 1°50'35"W 

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