Great Britain faced off with the United States, Poland, China, Germany, France, Jamaica and South Africa for a share of £1.5m at the new event on Saturday and Sunday night.
Bradshaw won the pole vault in 4.75m – a season’s best – and Meghan Beesley was second in the 400m hurdles in 55.90 seconds.
She said: “It’s a different event, it takes me back to my roots when I used to compete for Blackburn Harriers when you used to collect points.
“I kind of wanted to go over 1.82m which would have been a British record.
“But for me at the minute I am building nicely towards the European Championships.”
The USA won the event, and Great Britain came third.
America took the crown by 57 points ahead of Poland and Great Britain, who finished on 155 points – seven behind second spot – on Sunday.
GB would have claimed second had they not been disqualified from the men’s 4x400m – the final event – and finished ahead of Poland after a last-minute injury left them with just three runners.
The United States were the overnight leaders and comfortably took the crown, banking £370,000, which is split evenly between the squad, while Britain won around £264,000.
IAAF president Lord Coe hopes the format will eventually be a success, despite the sport’s top stars not taking part in the inaugural edition in London.
“I’ve been very clear that I want new things to be tried,” Coe said. “They are not always going to work out from the word go but that can’t inhibit us.”