"He'd have jumped out of his wheelchair," the 36-year-old, whose father has been paralysed since he broke his neck in a car accident in the south of France in 1986, told Reuters in an interview.
She said as much when addressing the factory last month after Williams announced the latest of a steady stream of promotions giving her ever more responsibility within the team.
When, more than a decade ago, the team's then-communications manager suggested she join the former champions as a press officer, her father's reaction almost set the furniture rattling.
"Dad said 'no way'," she smiled, recognising that there might have been an expletive in there as well.
"And quite rightly and I expected that. Twenty years ago, if Frank had thought I might be in this position, he'd have said 'no, absolutely no way'," she recalled.
Daughters may have a way of winning their fathers over but Claire has had to work harder than most to get Frank so convincingly onside.
"One of Williams' USPs (unique selling points) is family. I think it's important to retain that. And he'd never have put me in the job if he didn't think I could do it," she said.
"Other options haven't worked out so why not give it to a family member, at least you can trust them implicitly to have the best interests of the company at heart."
The nine times constructors' champions have always been central to her life, growing up with trips to Silverstone and barbecues on the lawn with the likes of Nigel Mansell.
Last weekend's Chinese Grand Prix in Shanghai, which Frank did not attend, was the first time in Formula One history that two of the 11 teams have been effectively run by women.
Williams was in charge, although her father was keeping a close eye on everything from his office back at the factory and Mike Coughlan was running the technical side, while Indian-born Monisha Kaltenborn is principal at Sauber.
Gender may have grabbed a few headlines, in a testosterone-fuelled sport without female race drivers, but there is no tokenism at a team that ranks among the most illustrious.
In Williams' case, the job has come to her after the last two designated successors to Frank - who has no immediate plans to step down despite celebrating his 71st birthday on Tuesday - abruptly departed.
Adam Parr resigned last year, after failing to see eye-to-eye with Formula One supremo Bernie Ecclestone, and Toto Wolff left in January after being unable to resist the challenge of leading Mercedes' motorsport activities.
"Frank is still our team principal and he founded the team. It's really important to Williams and all our employees, and I think to the sport, that Frank is still around," added Claire.
"But we are a business and we have to have some succession plans in place.
"If it turns out that it doesn't work out and I don't bring any improvements to the team and don't bring the budget in, then clearly I'm not the right person for the job and I'll step down.
"Williams is really important to me and I'm not about to let any of my ambitions get in the way of ensuring that Williams gets back to where we want it to get to."
Team bosses are not known for a lack of ambition or ego, but Williams - whose degree in politics will come in useful - was not exactly thrusting herself forward for the top job.
Asked at what point she began to think of eventually taking over from her father, the reply was immediate: "Never. I'm not one of those people at all. I was very happy being a press officer, I loved that job.
"I love working for Williams. It's never been about moving on up, making career progression. I never had a vision that I wanted to be the team principal and I never thought it would happen," she added.
"Up until a short time ago it wasn't really discussed. Working towards it or having a very clear goal wasn't the case. It was more that I want to do a really good job for the team in whatever capacity I am asked to do and will give it all I've got. The team is what is important."
Williams took on the new position only after consulting her two siblings and securing the blessing of her mother Virginia, who died of cancer last month.
"We are a really close family and I didn't want to do this without their full backing so I spoke to each and every one of them individually," she said. "I made sure that they were supportive of it, because this is their team as much as it is anybody else's. We all grew up with it."
Her job has not changed fundamentally, other than that she feels she has now "stuck my head above the parapet". Her focus is commercial and she will not sit on the pit wall during races because her father does not do that either.
What she will do is everything possible to pull Williams out of the prolonged slump they have fallen into.
Venezuelan Pastor Maldonado's win in Spain last year was the team's first since 2004 and they have yet to score a point in three races this season. They finished eighth overall last year.
"When you're winning it's great but it hurts more to lose than it feels great to win," said Williams. "And you do have to dig really deep. When it is a company that I've grown up in, I can't bear to see it like this.
"I can't bear to see that we've let down for whatever reason the people in the team that work so hard. I feel it is my responsibility to do whatever I can to turn it round."