Sorry they didn't honor your travel date. That really gets my back up.
You can apply for a transfer of vignette, but you'll need to wait until AFTER the vignette has expired, send an application, pay more money (a few hundred pounds), and wait several more weeks.
Personally, I would bite the bullet and travel to the UK together to collect the BRP. It *should* be at the post office but you may want to call to verify before booking. There are occasional times when the BRP isn't there yet, meaning your trip could be delayed on the return.
It will be fine that the kids aren't with you. IF (and I doubt it would happen) you were asked at the border, it's very easy to simply tell the truth. They didn't honor your travel date, so you are making a quick trip to collect the BRP, and return to the USA to tie up loose ends and will make the final trip with the kids in a few weeks.