You might lose flowers, but what's more likely is the flowers will be very late, so I wouldn't recommend it. You could cut back a bit after flowering, but B. davidii is one that should be hard pruned in late March, early April, and that pruning keeps them more controlled. Some of them do put on 10 feet of growth in a single season, but I expect you know that already;-))

If you prune now they will still flower but a bit later, well into August. But they will still try to grow up again, just as if you has pruned in March

You perhaps want to look at different cultivars that are less vigorous. Many of the best ones stay at about 2 metres, better than the new dwarf varieties that are a bit temperamental.

I'd say go ahead, just taking a few inches off.
If you garden with wildlife in mind, try to leave a couple unpruned this summer; those will flower early, the pruned ones will be a month to six weeks behind, feeding the butterflies over a longer period.